Fréchet Spaces

Andreas Kriegl
email:andreas.kriegl@univie.ac.at

442502, SS 2016, Mo.+Do. 1005 -1115 SR9

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0 /E / sN ×sN /E Q̃ / / Q̃ /0
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This is the script for my lecture course during the summer semester 2016. It
can be downloaded at http://www.mat.univie.ac.at/∼kriegl/Skripten/2016SS.pdf
Many of the proofs are taken from Meise and Vogt’s book [MV92] and I will give
detailed references to it, but also to Jarchow’s book [Jar81].
As prerequiste the user is assumed to be familiar with basic functional analysis
(for Banach spaces) and the basics of locally convex theory as presented in lecture
courses on higher functional analysis. I will refer to my script [Kri14] for these
results.
The main focus is on Fréchet spaces and additional topological properties for them.
Leading examples of Fréchet spaces will be the Köthe sequences spaces and in
particular the power series spaces with the space s of rapidly decreasing sequences
as most relevant member. We will have to consider several of these properties also
for general locally convex spaces, in particular, since the strong dual of Fréchet
spaces is rarely Fréchet.
Our discussion will start with properties of locally convex spaces which are pre-
served by the formation of inductive or projective limits. And we will then consider
what is inherited by the strong dual. Then we consider how properties of con-
tinuous linear maps translate into properties of the adjoint mappings using short
exact sequences. And we will introduce topological properties which garantee the
splitting of such sequences. These and further properties will also play a role in
determining situations where continuous linear mappings are locally bounded and
for characterizing the subspaces and the quotients of s.
I will put online a detailed list of the treated sections at the end of the semester
under http://www.mat.univie.ac.at/∼kriegl/LVA-2016-SS.html.
Obviously the attentive reader will find misprints and even errors. Thus I kindly
ask to inform me about such - future generations of students will appreciate the
corrections.

Andreas Kriegl, Vienna in February 2016

 Contents

1. Basics on Fréchet spaces 1

2. Colimit closed (coreflective) subcategories 17

Barrelled and bornological spaces 17

3. Limit closed (reflective) subcategories 23

Completeness, compactness and (DN) 23
Reflexive spaces 27
Montel spaces 28
Schwartz spaces 31
Tensor products 33
Operator ideals 40
Nuclear spaces 47

4. Duality 57
Spaces of (linear) functions 57
Completeness of dual spaces 58
Barrelledness and bornologicity of dual spaces 60
Duals of Fréchet spaces 64
Duals of Köthe sequence spaces 66
Semi-reflexivity and stronger conditions on dual spaces 68
Dual morphisms 84
Splitting sequences 99
Locally bounded linear mappings 116
The subspaces and the quotients of s 126

Bibliography 129

Index 131

andreas.kriegl@univie.ac.at c July 1, 2016 iii

 1. Basics on Fréchet spaces

In this section we describe ((reduced) projective) limits of locally convex spaces,

recall some basic facts on Fréchet spaces and introduce Köthe sequence spaces λp (A)
and in particular power series spaces λpr (A) as important examples.

1.1 Locally convex spaces

(See [Kri14, 1.4.4], [Jar81, 6.5 p.108], [MV92, 22 p.230]).
Let us recall that a locally convex space E is a linear space over the field
K ∈ {R, C} together with a compatible topology (i.e. addition E×E → E and scalar
multiplication K × E → E are continuous) and which has a 0-neighborhood basis
consisting of (absolutely) convex sets. Equivalently, the topology can be described
by a set P of seminorms (i.e. subadditiv and positive homogeneous functions p :
E → R). The correspondance is given by using the unit-balls {x : p(x) ≤ 1} of the
seminorms as 0-neighborhood subbasis and conversely considering the Minkowski-
functionals pU (see 1.3 ) for U in a 0-neighborhood basis consisting of absolutely
convex sets.
As usual we will require the topology to be Hausdorff or, equivalently, that the
−1
T
seminorms separate points, i.e. p∈P p (0) = {0}. We will abbreviate these
spaces by lcs.

1.2 Limits of lcs (See [MV92, 24 p.257], [Jar81, 2.6 p.37]).

Let F : (J, ) → lcs be a functor from a partially ordered set (or even a small
category) into the category of locally convex spaces, i.e. for every (object) j ∈ J we
are given an lcs F(j) and for every (morphism) j  j 0 a continuous linear mapping
F(j  j 0 ) : F(j) → F(j 0 ) satisfying F(j 0  j 00 ) ◦ F(j  j 0 ) = F(j  j 0  j 00 ).
Then the (inverse) limit of F is the lcs
n Y o
lim F := x = (xj )j∈J ∈ F(j) : F(j  j 0 )(xj ) = xj 0 for all j  j 0
j∈J

with the topology induced from the product

Q topology, i.e. the initial topology in-
duced by the projections prj : lim F ⊆ j∈J F(j) → F (j) for j ∈ J.
We call the limit a projective limit (and we write lim F instead of lim F), iff
←−
(J, ) is directed, i.e. ∀j1 , j2 ∈ J ∃j ∈ J : j  j1 , j2 .
If J 0 ⊆ J is initial in J, i.e. ∀j ∈ J ∃j 0 ∈ J : j 0  j, then lim F|JQ∼ = lim F: In
fact, the isomorphism is given by restricting the canonical projection j∈J F(j) →
0
Q
j 0 ∈J 0 F(j ) to the subspaces formed by the projective limits, see [Kri08, 3.13].

A projective limit is called reduced, iff all projections prj : lim F → F(j) have
←−
dense image. By replacing F(j) with the closure F̄(j) of the image of prj (lim F)
←−
in F(j) we get that lim F equals lim F̄, which is a reduced projective limit. Note
←− ←−
that F̄(j  j 0 ) is then a well defined continuous linear mapping with dense image.

andreas.kriegl@univie.ac.at c July 1, 2016 1

1.5

As closed subspace in the product the limit of complete lcs is complete. Recall,
that an lcs E is complete iff every Cauchy-net (i.e. x : (I, ) → E satisfying
∀p ∈ P ∀ε > 0 ∃i ∈ I ∀i0 , i00  i: p(xi0 − xi00 ) < ε) converges in E.

1.3 Complete lcs as limits of Banach spaces

(See [MV92, 24.5 p.260],[Kri14, 3.3.4]).
For absolutely convex A ⊆ E the Minkowski-functional pA is defined by

pA (x) := inf λ > 0 : x ∈ λA .
S
Note that {x : pA (x) < ∞} is the linear span hAivs = n∈N n A of A, which
coincides with E iff A is absorbing.
T The Minkowski-functional is a seminorm on
this subspace with kernel λ>0 λ A. If A is bounded (in each direction), then this
kernel is {0}. By EA we denote the resulting quotient space of hAivs , normed by
the norm induced by pA . If p is a seminorm on E and A := {x : p(x) < 1} its unit
ball, then we write Ep instead of EA = E/ ker(p). If A is absorbing we denote the
canonical quotient map ιA : E  EA and if A is bounded we denote the canonical
inclusion ιA : EA  E.
Every lcs E is a dense subspace of a (projective) limit of Banach spaces: For every
seminorm p we consider the completion Ẽp of the space Ep := E/ ker p, normed by
the uniquely determined seminorm p̃ : Ep → R with p = p̃◦pr : E → Ep → R. Then
Q Q
E embedds topologically into p Ep ,→ p Ẽp and in fact has dense image in the
(projective) limit (see [Kri08, 3.46]) where the connecting mappings ιpp0 : Ep → Ep0
for p ≥ p0 (and hence ker p ⊆ ker p0 ) are given by x + ker p 7→ x + ker p0 : In fact,
Q Q
let y ∈ limp Ẽp and U = p Up be a neighborhood of y in p Ẽp , i.e. Up = Ẽp
←−
for all but finitely many p1 , . . . , pn . Choose a p0  p1 , . . . , pn and a neighborhood
W of yp0 in Ẽp0 such that ιpp0i (W ) ⊆ Upi for 1 ≤ i ≤ n. Let x ∈ E be such that
ιp0 (x) ∈ W . Then ιpi (x) = ιpp0i (ιp0 (x)) ⊆ Upi for 1 ≤ i ≤ n, i.e. ι(x) ∈ U .
Thus, if E is complete, then it coincides with this limit. The limit is a (reduced)
projective one, since the set of seminorms of E can be assumed to be directed, i.e.
for each two seminorms p1 and p2 we may assume that max{p1 , p2 } is a seminorm
as well.

1.4 Lemma. Metrizable lcs

(See [Kri14, 3.5.2], [MV92, 25.1 p.276], [Jar81, 2.8.1 p.40]).
Let E be an lcs.
1. E has a countable 0-neighborhood basis.
⇔ 2. E has a countable basis of seminorms.
⇔ 3. The topology of E can be described by a translation invariant metric.

Proof. ( 3 ⇒ 1 ) The set {x : d(x, 0) < n1 } form a countable 0-neighborhood basis.

( 1 ⇒ 2 ) Take the Minkowski functionals of the 0-neighborhoods in the basis.
Q
( 2 ⇒ 3 ) Embed E ,→ n∈N Epn , a product of normed spaces. Then d(x, y) :=
P 1 kxn −yn k
n 2n 1+kxn −yn k gives the required metric.

1.5 Definition. Fréchet spaces

(See [Kri14, 2.2.1], [Jar81, 6.5.3 p.109], [MV92, 25.1 p.276]).
A Fréchet space ((F) for short) is a locally convex space, which satisfies the
equivalent conditions of 1.4 and is (sequentially) complete (equivalently, the trans-
lation invariant metric of 1.4.3 is complete).

2 andreas.kriegl@univie.ac.at c July 1, 2016

 1.8

Fréchet spaces are Baire spaces, hence the closed graph theorem (cf. [Kri14,
4.3.1], [MV92, 24.31 p.270], [Jar81, 5.4.1 p.92]) and the open mapping theorem
(cf. [Kri14, 4.3.5], [MV92, 24.30 p.270], [Jar81, 5.5.2 p.95]) hold for linear maps
between Fréchet spaces.

1.6 Remark. Equivalence of bases of seminorms.

Two sets P and P 0 of seminorms on a vector space E describe the same locally con-
vex space, iff each seminorm of one set is dominated by finitely
Pn many seminorms of
the other set (i.e. ∀p0 ∈ P 0 ∃n ∈ N ∃p1 , . . . , pn ∈ P: p0 ≤ i=1 pi , and conversely).
Thus for Fréchet spaces we may assume that we have an increasing sequence of
Pfact, we may replace a given countable set {pn : n ∈ N}
seminorms pn as basis: In
of seminorms by {p0n := i≤n pi : n ∈ N}.

1.7 Lemma. Stability of Fréchet spaces (See [MV92, 25.3 p.277]).

Closed subspaces of Fréchet spaces are Fréchet and quotients of Fréchet spaces by
closed subspaces are Fréchet. Limits of countable many Fréchet spaces are Fréchet.
The Fréchet spaces are exactly the (projective) limits of sequences of Banach spaces.

Proof. The trace of the countable 0-neighborhoodbasis (or countable many semi-
norms) is a 0-neighborhoodbasis (are the generating seminorms) of the subspace.
The quotient seminorms q̃(x + F ) := inf{y ∈ F : q(x + y)} are a basis of semi-
norms on the quotient, see [Kri14, 3.3.3]. And since Cauchy-sequences can be lifted
along the quotient mapping (see [Kri14, 3.5.3]) the quotient is (sequentially-)com-
plete as well.
We obviously get a countable basis of seminorms for the product of countable many
Fréchet spaces, and since limits of complete spaces are complete, such a limit is a
Fréchet space.

1.8 Examples of Fréchet spaces.

1. `p , c0 : Every Banach space (in particular, `p for 1 ≤ p ≤ ∞ and c0 ) is a

Fréchet space.
2. KN : The space KN of all sequences is a Fréchet space with respect to the
product topology, i.e. the pointwise(=coordinatewise) convergence. It is the
N
P of K for n ∈ N, in fact Epn = K , when E := K and pn (x) :=
n n
limit
i<n |xi |.
3. C(X): Let C(X, K) be the space of continuous functions on a topological
space X supplied with the topology of uniform convergence on the compact
subsets K ⊆ X, i.e. induced by the seminorms pK : f 7→ kf |K k∞ . In
order for C(X, K) to be complete we need, that a function, with continuous
restrictions on all compact subsets, is continuous. This is the case, when X is
a Kelley-space (i.e. carries the final topology with respect to its compact
subsets). Then C(X, K) = limK C(K, K), since C(X, K)pK = C(X, K)/{f :
←−
f |K = 0} = C(K, K). If X has a countable basis for the compact sets, then
C(X, K) is metrizable.
4. H(U ): If U ⊆ Cn is open, then the space H(U ) of holomorphic functions
on U is a closed subspace of C(U, C), hence Fréchet.
5. C ∞ (X): Let X be an open subset of some Rn or a smooth finite dimensional
paracompact connected manifold. Then the space C ∞ (X, K) of smooth func-
tions on X is a Fréchet space with the topology of uniform convergence of
each derivative separately on compact subsets (contained in some chart).

andreas.kriegl@univie.ac.at c July 1, 2016 3

1.11

6. S: The space S of rapidly decreasing functions on Rn is a Fréchet

space, where the seminorms are given by sup{(1 + kxk)j kf (k) (x)k : x ∈ Rn }
for j, k ∈ N.
7. CW (U ): Spaces of weighted continuous functions. Let X be a Kelley-
space and W a (countable) set of non-negative upper semi-continuous
(i.e. w−1 ([α, ∞)) is closed for all α ∈ R) functions w : X → R. Then
CW (U, K) := {f ∈ C(U, K) : w · f is bounded for each w ∈ W}, cf. [Sch12,
4.3 p.76]. A particular case is 3 , where W = {χK : K ⊆ U is compact}, cf.
[Sch12, 4.4 p.76].
8. HW (U ): Spaces of weighted holomorphic functions. Let U ⊆ C be open
and W be as in 7 . Then HW (U ) := H(U, C) ∩ CW (U, C). [Sch12, 4.3 p.76]
9. C (M ) (U ): Let U ⊆ Rn be open and Mk be a sequence of positive real
numbers. The space of Denjoy-Carleman functions on U of Beurling
type is
 n kf (j) (x)k
C (M ) (U, K) := f ∈ C ∞ (U, K) :kf kK,ρ := sup : j ∈ N, x ∈ K < ∞
o
j! Mj ρ j

for all compact K ⊆ U and ρ > 0 .

1.9 Definition. Köthe sequence spaces

(See [MV92, 27 p.307], [Jar81, 1.7.E p.27]).
Let A be a set of R-valued sequences, which satisfies ∀n ∈ N ∃a ∈ A: an 6= 0.
Then for 1 ≤ q ≤ ∞ the Köthe sequence space λq (A) is defined as
λq (A) := x ∈ KN : ∀a ∈ A : a · x ∈ `q


with the seminorms given by x 7→ kxka := ka · xk`q . Moreover,

c0 (A) := x ∈ λ∞ (A) : ∀a ∈ A : x · a ∈ c0


as subspace of λ∞ (A).

1.10 Remark.

1. We may (and will always) assume that all a ∈ A are R+ : {t ∈ R; T ≥ 0}-

valued, since obviously λp (A) = λp (|A|), where |A| := {j 7→ |aj | : a ∈ A}.
2. We may (and will always) assume that A is directed, i.e.
∀a, b ∈ A ∃c ∈ A ∀n ∈ N : cn ≥ max{an , bn } :
P
Otherwise, let à := { a∈ã a : ã ⊆ A finite}. Then à ⊇ A is directed and
P P P
kxkã := k a∈ã a · xk`p ≤ a∈ã ka · xk`p =: a∈ã kxka . Now apply 1.6 .
3. If A is countable, we may replace A by an increasing sequence {ãn : n ∈ N}:
In fact, let A = {an : n ∈ N} and ãn :=
P
k≤n k . Then kan · xkp ≤
a
P P
kãn · xkp = k k≤n ak · xkp ≤ k≤n kak · xkp , cf. 1.6 .

1.11 Lemma. Köthe sequence spaces as limits (See [MV92, 27.2 p.307]).
The Köthe sequence space λq (A) is isomorphic to lim F, where the functor F on
0
(A, ≥) is given by F(a) := `q and F(a ≥ a0 ) : F(a) → F(a0 ) is given by x 7→ aa x,
where ( 0
an
a0 for an 6= 0,
: n 7→ an
a 0 for an = 0 (and hence a0n = 0).

4 andreas.kriegl@univie.ac.at c July 1, 2016

 1.13

In particular, if A is countable, then λq (A) and c0 (A) are Fréchet spaces (See
[MV92, 27.1 p.307]).

Proof. The isomorphism is given by λq (A) 3 x 7→ (a · x)a∈A with inverse mapping

a(n)
1
lim F 3 y 7→ x := ( a(n) n
yn )n∈N , where the a(n) ∈ A are choosen such that
a(n)n > 0.
( ) Let y = (y a )a∈A ∈ lim F ⊆ a∈A `q be given. For b ∈ A and n ∈ N let
Q


c(n) ≥ max{a(n), b}. Then y b = c(n) b

y c(n) and y a(n) = a(n)
c(n) y
c(n)
, thus (since
c(n)n ≥ a(n)n > 0):
bn c(n) bn c(n)n a(n) bn a(n)
ynb = yn = yn = y = (b · x)n .
c(n)n c(n)n a(n)n a(n)n n

1.12 Convention. Calculating with ∞.

Put ∞ ≥ x ∀x, 0 + ∞ := ∞, 0 · ∞ := 0 and extend + and · by monotonicity and
commutativity to mappings [0, ∞] × [0, ∞] → [0, ∞]. Then
∞ ≥ x + ∞ ≥ 0 + ∞ = ∞ ⇒ ∀x ≥ 0 : x + ∞ = ∞,
y
∀0 < x, y < ∞ : x · ∞ ≥ x · = y ⇒ ∀x > 0 : x · ∞ = ∞,
x
and then + and · are associative and distributiv.
Let 1/0 := ∞, 1/∞ := 0 and x/y := x · y1 . Then
(
1 0 for x = 0 1
x/0 := x · = x · ∞ := and x/∞ := x · = x · 0 = 0.
0 ∞ for 0 < x ≤ ∞ ∞

1.13 Remark. Köthe sequence spaces as reduced projective limits.

Let E = λp (A) (resp. E = c0 (A)) and ` = `p (resp. ` = c0 ). For a = (ak )k ∈ A the
mapping x 7→ a · x, E → ` and hence x 7→ ka · xk`p =: kxka has kernel
ker k ka = {x ∈ E : x|Na = 0}, where Na := carr a := {k : ak 6= 0}.
By assumption on Köthe sequence spaces N = a∈A Na . Define the Banach spaces
S

`p (a) := {x ∈ RNa : ka · xk`p < ∞} ∼

= `p (carr a) := {x ∈ `p : carr x ⊆ carr a}
c0 (a) := {x ∈ `∞ (a) ⊆ RNa ⊆ RN : lim a · x = 0}

Obviously, the coproduct R(Na ) is dense in `p (a) for p < ∞ and hence also R(N) ⊆ E,
since R(N) ⊆ E/ ker k ka ⊆ `p (a). ⇒ Ea := (E/ ker k ka )∼ ∼ = `p (a) for 1 ≤ p < ∞
∼
(resp. Ea = c0 (a)). By completeness E = lima Ea .
←−
Ea ∼= `p (a) ∼
·a / `p (carr a)   / `p
77 =
|carr a

E = λp (A) |carr a0
0
· aa · aa
0

|carr a0 ''   
E a0
∼
= `p (a0 )
·a0 / `p (carr a0 )   / `p
∼
=

andreas.kriegl@univie.ac.at c July 1, 2016 5

1.15

For p = ∞ however, we only get c0 (a) ⊆ Ea ⊆ `∞ (a) and not necessarily Ea =

`∞ (a), e.g. for E := s, see . Nevertheless
\
lim Ea = λ∞ (A) = `∞ (a) = lim `∞ (a),
← −
a
← −
a
a
but the projective limit on the right side is not reduced!

1.14 Definition. Power series space (See [MV92, 29 p.337]).

A particular case of Köthe sequence spaces is, when A = Aα,r := {j 7→ etαj : t < r}
for some r ∈ R and a fixed sequence (αj )j increasing monotone towards +∞. Then
λqr (α) := λq (Aα,r ) is called power series space (of finite type if r < +∞ and of
infinite type if r = +∞). Note that for r < ∞ the mapping Φ : λqr (α) → λq0 (α),
x 7→ (er αj xj )j is an isomorphism, since kΦxkt = kxkt+r , see 1.26.1 .

1.15 Examples of Köthe sequence spaces (See [MV92, 29.4 p.339]).

∼ `p and c0 (A) =
1. If A is a singleton, then λp (A) = ∼ c0 .
2. Let A := {en : n ∈ N}, where en are the standard unit vectors in RN . Then
λ∞ (A) = λp (A) = c0 (A) = RN for all p ∈ [1, ∞]. Note that we can equally
take {χF = max{ek : k ∈ F } : F ⊆ N is finite} instead of A.
Let A = RN be the set of all real sequences (ak )k . Then λ∞ (A) = K(N) :=
3. `
j∈N K (cf. [Kri14, 3.6.1]): Suppose there is an x ∈ λ (A) with carr(x)
∞

being not finite. Now define a ∈ A as ak := k/|xk |, which should be (say) 1 if

xk = 0. Then |(a·x)k | = k for all k ∈ carr x, hence
P is not bounded. A basis of
seminorms on the coproduct is given by x 7→ k |ak xk | ≤ 2 sup{|2k ak xk | :
k ∈ N}, with ak ∈ R. This space is not Fréchet!
4. Let A be the set of all polynomials. Then s := λ∞ (A) is the space of
fast falling sequences. We get the same space if we use the subset
{n 7→ nk : k ∈ N} ⊆ A or better {n 7→ (1 + n)k : k ∈ N} instead of A, since
this sequence is increasing. Note that we should put 00 := 1 (otherwise,
the first set will not satisfy the requirements for a Köthe sequence space)
but thenPthe set is not linearly ordered (since 00 > 0k for k > 0). Let
k
P
p : x 7→ k≤d ak x . Then k kp ≤ k≤d ak k kk .
1
Moreover, kn 7→ (1 + n)k xn k`1 ≤ kn 7→ (1 + n)k+2 xn k`∞ · n (1+n)
P
2 , hence
p
s = λ (A) = c0 (A) for all 1 ≤ p ≤ ∞.
The space s is the power series space λ∞ (α) for α(n) := ln(1 + n).
5. If A = {n 7→ rn : r > 0} = {n 7→ es n : s ∈ R} then λ∞ (A) = λ1 (A) = H(C),
the space of entire functions. It is the power series space λ∞ (α) for
α(n) := n (See [MV92, 29.4.2 p.340]).
 X∞ 
x 7→ z 7→ xn z n
n=0

In fact, the power series n an z converges for all |z| < R iff {an rn : n ∈ N}
n
P
is bounded (equivalently, absolutely summable) for all r < R.
6. If A = {n 7→ rn : 0 < r < 1} = {n 7→ es n : s < 0} then λ∞ (A) = λ1 (A) =
H(D), the space of holomorphic functions on the unit disk [MV92,
29.4.3 p.340].
 X∞ 
x 7→ z 7→ xn z n
n=0

It is the power series space λ0 (α) for α(n) := n (See [MV92, 29.4.2 p.340])

6 andreas.kriegl@univie.ac.at c July 1, 2016

 1.16

1 1
7. For 1 ≤ p < ∞ and q + p = 1 we have λ1 (`p ) = (`q , σ(`q , `p ) as lcs:
1
(⊇) By the Hölder inequality kxkλy = kx · yk`1 ≤ kxk`q · kyk`p < ∞ for all
y ∈ `p and x ∈ `q .
(⊆) Let x ∈ KN be such that kx·yk1 < ∞ for all y ∈ `p . Then the linear map
y 7→ Px · y, `p → `1 has closed graph and thus is continuous. Consequently,
y 7→ n xn · yn is a continuous linear functional, hence x ∈ (`p )∗ = `q (see
[Kri14, 5.3.1]).
λ1 (`∞ ) = (`1 , σ(`1 , `∞ )): For (⊆) choose y = 1.
λ1 (c0 ) = (`1 , σ(`P
1
, c0 )): Suppose x ∈ λ1 (c0 ) \ `1 , choose k 7→ nk strictly
nk+1
increasing with j=n k +1
|xj | ≥ k and yj := k1 for nn < j ≤ nk+1 . Then
P
kx · yk`1 ≥ k 1 = ∞.

1.16 Proposition. Function spaces isomorphic to s

(See [MV92, 29.5 p.340]).
∞
The following spaces are isomorphic to s: C2π (R), S(R), C[a,b]
∞
(R), and C ∞ ([a, b]).

(R) ∼
 R 
∞ 1 π −ikt
Proof. (1) C2π = s via Fourier-coefficients f 7→ 2π f (t) e dt ,
Rπ
−π k∈Z
cf. [Kri07b, 5.4.5] and 1.26.3 : Let ck (f ) := 2π −π f (t) e−ikt dt. Then ck (f 0 ) =
1

P[Kri07b, 5.4.4 p.101] or [Kri06, 9.3.5], f ∈ L ⇒ (ck (f ))k∈Z ∈ c0 and

1
ik ck (f ) by
1
c ∈ ` ⇒ k ck expk converges absolute in C by Riemann-Lebesgue [Kri07b, 5.4.1
p.95], [Kri06, 9.3.6]. Note, that that s is taken with index set Z instead of N, but
see 1.26.3 .
(2) S(R) ∼= s (See [MV92, 29.5.2 p.341]):
Let ρ(t) := e−t and consider the Hilbert space completion L2ρ (R) of the space of
2

R
polynomials with respect to the inner product hf |giρ := R f (t)g(t) ρ(t) dt. Ob-
viously L2ρ (R) ∼
= L2 (R) via f 7→ ρ f . Gram-Schmidt orthonormalization applied
√
to the monomials t 7→ tn gives an orthonormal basis ( √ n1 √ Hn )n∈N , where Hn
2 n! π
are the Hermite polynomials (cf. [Kri07b, 6.3.9 p.118]), which can also be
obtained recursively H0 := 1, Hn+1 (t) := 2t Hn (t) − 2n Hn−1 (t):
From the recursion we get Hn0 = 2n Hn−1 by induction. In fact H00 = 0, H1 (t) = 2t,
H10 = 2H0 , and hence
0

0
Hn+1 = 2 id Hn − 2n Hn−1 = 2 Hn + 2 id Hn0 − 2n Hn−1 0

= 2 Hn + 4n id Hn−1 − 4n(n − 1) Hn−2

= 2 Hn + 2n · (2 id Hn−1 − 2(n − 1) Hn−2 ) = 2(n + 1) Hn
(n)
Moreover, Hn = (−1)n ρ ρ since
(n)
 (n)
0
nρ nρ
Hn+1 = 2 id Hn − Hn0 = 2 id (−1) − (−1)
ρ ρ
(n) (n+1) (n) 
 ρ ρρ − (−2 id ρ) ρ ρ(n+1)
= (−1)n 2 id − = (−1)n+1 .
ρ ρ2 ρ
By induction we get for m ≥ n:
Z Z Z
0
hHm+1 |Hn iρ = ρ Hm+1 Hn = ρ (2 id Hm − Hm ) Hn = (−ρ0 Hm − ρ Hm
0
) Hn
Z Z Z
= ρ Hm Hn0 − (ρ Hm Hn )0 = 2n ρ Hm Hn−1 = 0.

andreas.kriegl@univie.ac.at c July 1, 2016 7

1.16
R √
Finally, ρ= π and again by induction
Z Z Z
part.int.
kHn k2ρ = ρ Hn2 = (−1)n Hn ρ(n) = ====== = (−1)n−1 Hn0 ρ(n−1)
Z
= 2n Hn−1 (−1)n−1 ρ(n−1) = 2n kHn−1 k2ρ
√ √
= 2n π 2n−1 (n − 1)! = 2n n! π.
√
ρ
Thus the corresponding Hermite functions hn := √ √ Hn form an ortho-
2n n! π
normal basis of L2 (R). For A± : S → S, defined by f 7→ id ·f ∓ f 0 , we have:
√ √
( ρ Hn )0 − id ρ Hn
A− (hn ) := h0n
+ id hn = p √
2n n! π
1 √ 2n √
=p √ ρ Hn0 = p √ ρ Hn−1
n
2 n! π n
2 n! π
√
2n √ √
= p √ ρ Hn−1 = 2n hn−1
2n−1 (n − 1)! π
∞ ∞
onb
X part.int. X
⇒ Am+ f ==== hA m
+ f |hn i hn =
==== = =
= hf |Am
− hn i hn
n=0 n=0
X p
m/2
= 2 n(n − 1) . . . (n − m + 1) hf |hn−m i hn
n≥m
onb
=
==⇒ |hAm 2 m
+ f |hn+m i| = 2 (n + m)(n + m − 1) . . . (n + 1)|hf |hn i|
2

∞
X ∞
X
⇒ nm |hf |hn i|2 ≤ 2−m |hAm 2
+ f |hn+m i| ≤ 2
−m
kAm 2
+ f kL2 (R) < ∞,
n=0 n=0
hence S → s, f 7→ (hf |hn i)n≥0 isPcontinuous and obviously injective.
It is also onto: Let a ∈ s. Then n an hn converges in S, since
X √ p
− h0k + id ·hk = A1+ (hk ) = 21/2 n hhk |hn−1 i hn = 2(k + 1) hk+1
n≥1
r r r r
n n+1 n n+1
⇒ h0n = hn−1 − hn+1 and id ·hn = hn−1 + hn+1 .
2 2 2 2
∞
(R) := f ∈ C ∞ (R) : f (x) = 0 ∀x ∈
/ [a, b] ∼

(3) C[a,b] =s
(See [MV92, 29.5.3 p.342]):
W.l.o.g. −a = b = π/2.
Φ : S(R) → C[−π/2,π/2]
∞
(R), Φ(f )(t) := f (tan(t)) ∀|t| < π/2 is an iso, since
p
(R, R)
X g̃j,p ∞
Φ(f )(p) = j+p
f (j) ◦ tan with g̃j,p ∈ C2π
j=1
cos

| tan(t)k f (j) (tan(t))| ≤ sup |xk f (j) (x)| =: Ck,j < ∞ ∀|t| < π/2
x∈R
1
Since tan(x) ∼ for x near ±π/2 we get |Φ(f )(p) (t)| → 0 for t → ±π/2.
cos(x)
And the inverse mapping is given by f 7→ f ◦ arctan using analogous arguments:
qn (s)
arctan0 (s) = 1+s
1
2 ⇒ arctan
(n)
(s) = (1+s 2 )n with deg(qn ) ≤ n − 1. Thus

p
X tk gj,p (t) (j)
tk Φ−1 (f )(p) (t) = f (arctan(t)) with deg(gj,p ) ≤ n − 1,
j=1
(1 + t2 )n

8 andreas.kriegl@univie.ac.at c July 1, 2016

 1.18

and
 n+k−1  
tn+k−1 f (j) (arctan(t)) = tan ±(π/2 − s) f (j) ±(π/2 − s)
 
= (± cot(s))n+k−1 f (j) ±(π/2 − s)
 n+k−1 (j)
±s cos(s) f (±(π/2 − s))
= → 0 for s & 0.
sin(s) sn+k−1
Now the result follows since S(R) ∼
= s by 2 .
∞ ∼ s (See [MV92, 29.5.4 p.343]):
(4) C ([a, b]) =
W.l.o.g. −a = b = 1.
Φ : f 7→ f ◦ cos, C ∞ ([−1, 1]) → C2π,even
∞ ∼
=s
is continuous and injective. It is also onto, since
g0 g0
f := g ◦ arccos ∈ C([−1, 1]) ∩ C ∞ (]−1, 1[), f 0 = − ◦ arccos, and ∞
∈ C2π,even .
sin sin
∞ ∞
Note that via Fourier-coefficents C2π,even = {f ∈ C2π : f (x) = f (−x)} = {f ∈
∞ ∼ P

C2π : cn (f ) = c−n (f )} = s, via x 7→ n≥0 an cos(n x) ← (an )n . Thus s →
C ∞ ([−1, 1]) is given by a 7→
P P
n∈N an cos(n arccos t) = n∈N an Tn , where Tn :
t 7→ cos(n arccos t) are the Tschebyscheff(=Chebyshev) polynomials.

1.17 Definition. Schauder-basis and absolute basis.

A sequence (ej )j∈N is called Schauder-basis in [MV92, Def. in 24.27 p.322] (or
called topological basis in [Jar81, 14.2 p.292]) of the lcs E, if
∀x ∈ E ∃! ξ = (ξj (x))j ∈ KN : x =
X
ξj (x) ej .
j

The mappings x 7→ ξj (x) are then linear.

Obviously, the standard basis (ej )j∈N is a Schauder-basis in λp (A) for any A:
 n
X  X X 1/p
a· x− xj ej = aj xj ej = |aj xj |p → 0.
`p `p
j=0 j>n j>n

A Schauder-basis is called absolute basis (See [MV92, Def. in 24.27 p.322],

[Jar81, 14.7.6 p.314]), iff
X
∀p ∃p0 ∃C > 0 ∀x : |ξj (x)| p(ej ) ≤ C p0 (x)
j

The standard basis is an absolute basis in λp (A) iff λp (A) = λ1 (A):

X
∀a ∃a0 ∃C > 0 ∀x : |xj | kej · ak`p ≤ C kx · a0 k`p ,
j
| {z }
=|aj |

i.e. kx · ak`p ≤ kx · ak`1 ≤ C kx · a0 k`p .

1.18 Lemma on (F) with Schauder-basis (See [MV92, 28.10 p.331]).

Let F be a Fréchet-space with Schauder-basis (ej )j and corresponding coefficient
functionals ξj . Then
X 
∀p ∃p0 ∃C ∀x : sup p ξj (x)ej ≤ C p0 (x).
k∈N
j≤k

andreas.kriegl@univie.ac.at c July 1, 2016 9

1.21

Proof. Let (k kn )n be an increasing basis of seminorms of F . We consider

Pk
new seminorms p̃n (x) := supk∈N j=1 ξj (x) ej n . Obviously, k kn ≤ p̃n since
P
j≤k ξj (x)ej converges to x, thus the metrizable locally convex topology τ in-
duced by the seminorms p̃n is finer than the given one. In order to apply the
open mapping theorem it is enough to show completeness of τ : Let (xm )m be a
Cauchy-sequence for τ . We have
k k−1
0 00 X 0 00 X 0 00
ξk (xm ) − ξk (xm ) kek kn ≤ ξj (xm − xm ) ej + ξj (xm − xm ) ej
n n
j=1 j=1
0 00
≤ 2 p̃n (xm − xm ).
Since ∀k ∃n : kek kn > 0 the sequence (ξk (xm ))m is Cauchy in K, let x∞
k be its limit.
Since (xm )m is Cauchy, we have

∀n ∀ε > 0 ∃m ∀m0 , m00 ≥ m ∀k :

k k
0 00 X 0 X 00
ε ≥ p̃n (xm − xm ) ≥ ξj (xm ) ej − ξj (xm ) ej .
n
j=1 j=1

With m00 → ∞ we obtain

k k
X 0 X
ξj (xm ) ej − x∞
j ej ≤ ε.
n
j=1 j=1

Thus
k+p
X k+p
X
∀k, p : x∞
j ej ≤ 2ε + ξj (xm ) ej .
n n
j=k+1 j=k+1
Since j ξj (xm ) ej converges in E, the sequence j x∞
P P
j ej is Cauchy, hence con-
∞
verges to some x∞ := j=0 x∞ ∞ ∞
P
j e j ∈ E with ξ j (x ) = xj , since (ej ) is a Schauder-
0
m ∞
basis. By the inequality above, we have that x → x with respect to τ .

1.19 Corollary. Schauder-bases in (F) have continuous coefficients

(See [MV92, 28.11 p.332]).
Let F be a Fréchet-space with Schauder-basis (ej )j and corresponding coefficient
functionals ξj . Then ∀p ∃p0 ∃C > 0 ∀x ∀j : |ξj (x)| p(ej ) ≤ C p0 (x).
In [Jar81, 14.2 p.292] a Schauder-basis is defined as a topologogical basis for which
the coefficient functionals are continuous.

1.20 H(DR ) has (z k )k∈N as absolute basis (See [MV92, 27.27 p.323]).
Let DR := {z ∈ C : |z| < R} be the disk with radius 0 < R ≤ ∞. Taylor
P (k)
development f (z) = k f k!(0) z k shows that (z 7→ z k )k∈N is a Schauder-basis of
H(DR ). This is even an absolute basis: kf kr := sup{|f (z)| : |z| ≤ r} for r < R is a
basis of seminorms and ∀f ∈ H(DR ) ∀r < r0 < R :
X f (j) (0) Z ∞  j
[Kri11, 3.30] X 1 f (z) X r
kz j kr ========== dz r j
= kf kr0 .
j
j! j
2πi |z|=r0 z j+1
j=0
r0

1.21 The Fréchet spaces with absolute basis are the spaces λ1 (A)
(See [MV92, 27.26 p.323], [Jar81, 14.7.8 p.314]).
For Fréchet space E we have: ∃A countable: E ∼
= λ1 (A) ⇔ E has an absolute basis.

Proof. (⇒) The standard basis (ej )j∈N is obviously an absolute basis of λ1 (A).

10 andreas.kriegl@univie.ac.at c July 1, 2016

 1.23

(⇐) (See [MV92, 27.25 p.322]) Let (ej )j be an absolute basis of E and consider
the Köthe matrix A := (j 7→ kej kp )p∈N . Then ξ : E → KN , x 7→ (ξj (x))j is linear.
(ej )j absolute basis ⇒
X
∀p ∃p0 ∃C ∀x : |ξj (x)| kej kp ≤ Ckxkp0
j
1 1
⇒ ξ(x) ∈ λ (A) and ξ : E → λ (A) continuous and injective.
Claim: ξ is onto λ1 (A):
n+k
X n+k
X
y = (yj )j ∈ λ1 (A) ⇒ yj ej ≤ |yj | kej kp
p
j=n+1 j=n+1

y ∈ λ1 (A) P P
= ⇒ n 7→
======= j≤n yj ej Cauchy in E ⇒ converges to x := j yj ej with ξ(x) = y,
open map.thm.
i.e. ξ onto. = ⇒ ξ is isomorphism.
===========

1.22 Dual space of λp (A) (See [MV92, 27.11 p.313]).

Let λ := λp (A) with 1 ≤ p < ∞ or λ := c0 (A). Then

x∗ 7→ (x∗ (ej ))j∈N , λ∗ → λ1 (λ) := y ∈ KN : ∀x ∈ λ :

n X o
|xj yj | < ∞
j

is linear and injective. If A is countable it is even bijective.

Proof.
() y ∈ λ∗ :
∞
X ∞
X  X ∞
∀x ∈ λ : x = xj ej ⇒ y(x) = y xj ej = xj y(ej ) .
j=0 j=0 j=0
| {z }
=:yj
X
kεk∞ ≤ 1 ⇒ ε · x ∈ λ, hence xj yj converges absolutely.
j

() y ∈ λ1 (λ) ⇒ y n := χ{1,...,n} · y ∈ λ∗ and

X ∞
X
lim y n (x) = lim xj yj = xj yj =: y(x) ∀x ∈ λ
n→∞ n→∞
j≤n j=0

Un := {x ∈ λ : |y n (x)| ≤ 1} ⇒ U := n∈N Un is barrel (see 2.1 ), y ∈ U o , λ

T

barrelled by 2.3 ⇒ U 0-nbhd, hence y ∈ λ∗ .

Counter-example.
Let A = c0 . By 1.15.7 we have λ := λ1 (A) = (`1 , σ(`1 , A)), and hence λ∗ = A =
c0 , whereas λ1 (λ) = λ1 (`1 ) = `∞ .

1.23 Minkowski-functionals on polars in the dual.

For any subset A ⊆ E we have the polar
Ao := {x∗ ∈ E ∗ : |x∗ (x)| ≤ 1 ∀x ∈ A}.
The Minkowski-functional pAo (on the linear span of Ao ) is given by
n o n x∗ o
pAo (x∗ ) := inf λ > 0 : x∗ ∈ λ Ao = inf λ > 0 : (x) ≤ 1 ∀x ∈ A
λ
n o n o
= inf λ > 0 : |x (x)| ≤ λ ∀x ∈ A = sup |x∗ (x)| : x ∈ A = kx∗ |A k∞ .
∗

andreas.kriegl@univie.ac.at c July 1, 2016 11

1.25

In the particular case, where A = U ⊆ E is a 0-neighborhood, the polar U o is

bounded in the strong dual E ∗ . In fact, the strong topology is that of uniform
convergence on the bounded sets B ⊆ U , i.e. given by the seminorms k |B k∞ .
Since B is bounded, it is contained in K · U for some K > 0, hence kx∗ |B k∞ :=
sup{|x∗ (x)| : x ∈ B} ≤ sup{|x∗ (K u)| : u ∈ U } ≤ K sup{|x∗ (x)| : x ∈ U }, which is
at most K for x∗ ∈ U o .

1.24 Minkowski-functionals for polars of 0-nbhds in λp (A)

(See [MV92, 27.12 p.313]).
Let λ := λp (A) for 1 ≤ p < ∞ or λ := c0 (A). For a ∈ A let Ua := {x ∈ λ :
ka · xk`p < 1} and k k∗a := p(Ua )o = k |Ua k∞ with unit-ball (Ua )o . Then
y 1 1
kyk∗a = for + = 1 or q = 1 in case λ = c0 (A).
a `q p q

Proof. Let first 1 < p < ∞ and y ∈ λ∗ . We assume first, that carr y ⊆ carr a.
Then
1.23 1.22 X X
kyk∗a =
===== ky|Ua k∞ = sup |y(x)| =
===== sup xj yj = sup xj yj
j∈N
x∈Ua x∈Ua x∈Ua j∈carr a
n X yj o `q = (`p )∗ y
= sup xj aj · : (xj aj )j∈carr a `p
≤ 1 ======== ,
j∈carr a
aj a `q

and for carr y 6⊆ carr a we get ∞ on both sides.

Analogous for λ = λ1 (A) and λ = c0 (A).

1.25 Theorem. Equality of λp (A) for various p (See [MV92, 27.16 p.315]).

0
1. ∃ 1 ≤ p 6= p0 ≤ ∞: λp (A) = λp (A) as lcs;
0
⇔ 2. ∀ 1 ≤ p 6= p0 ≤ ∞: λp (A) = λp (A) as lcs;
⇔ 3. ∀a ∈ A ∃a0 ∈ A: a
a0 ∈ `1 .

If A is countable, then it is enough to assume equality in 1 and 2 only as sets.

0
Proof. If A is countable, and p < p0 ⇒ λp (A) → λp (A) continuous injective
open map.thm. 0
= ⇒ λp (A) = λp (A) as Fréchet spaces in 1 and 2 .
===========
0
( 3 ⇒ 2 ) Since λp (A) → λp (A) injects continuously for 1 ≤ p ≤ p0 ≤ ∞, we have
to show that λ∞ (A) injects continuously in λ1 (A). Let a ∈ A. ∃a0 satisfying 3 .
⇒
1 X X aj λ∞ (A) a
∀x ∈ λ∞ (A) : kxkλa (A) := |xj aj | = xj a0j 0 ≤ kxka0 .
j j
aj a 0 `1

( 2 ⇒ 1 ) is trivial.
( 1 ⇒ 3 ) For p0 = ∞ we get λ∞ (A) = c0 (A), since `p ⊆ c0 ⊆ `∞ .
0
p λp (A)
⇒ ∀a ∃a0 ≥ a ∃C > 0 : k kλa (A)
≤ C k ka0
[Kri14, 1.3.3,1.3.7] 0 p0
⇒ ∀a ∃a0 ≥ a ∃C > 0 : Uap0 := {x : kxkλa0 ≤ 1} ⊆ C Uap
===============
1.24
⇒ ∀y ∈ h(Uap )o ilin.sp : kyk∗a0 ,p0 ≤ Ckyk∗a,p .


=
====

12 andreas.kriegl@univie.ac.at c July 1, 2016

 1.26

1/q + 1/p := 1; 1/q 0 + 1/p0 := 1. ∀η ∈ `q , kηk`q ≤ 1

Hölder, 1.22 0
==========
= ⇒η · a ∈ (Uap )o ⊆ C (Uap0 )o , since ka · xk`p ≤ 1 ∀x ∈ Uap .
1.24 X  0
0 aj
q 0 1/q
====
= ⇒ |ηj |q 0 = kη · a|U p0 k∞ ≤ C
j
aj a0

0
ξ := η q
nX aj
q0 0 0
o 0
====== ⇒ sup |ξj | 0 : ξ ∈ `q/q , kξk`q/q0 = (kηk`q )q ≤ 1 ≤ C q
j
aj

q q0 0
t := q−q 0 , i.e. 1
t + q = 1, (`q/q )∗ = `t

 a q 0 X aj q0 t   a q0 t
1.15.7 0
t
⇒ 0
====== ∈ ` and = ≤ C q t.
a j
a0j a0 `t

⇒ ∃d := q 0 t ≥ 1 ∀a ∃a0 < ∞. W.l.o.g. d ∈ N. Let a(0) = a and choose

0 d
P
j (aj /aj )
P (k) (k+1) d
a(1) , a(2) , . . . , a(d) recursively with j (aj /aj ) < ∞ for 0 ≤ k < d.

Hölder inductive
X a(0)
j
Y a(k)
X d−1 j
⇒
============= (d)
= (k+1)
< ∞.
j aj j k=0 aj

1.26 Proposition. Equalities for power series spaces

(See [MV92, Aufgabe 1+2 p.323]).

1. Let 0 < α = (αn )n % ∞, R ∈ [0, ∞), p ∈ [1, ∞]. Then λpR (α) ∼ p
= λ0 (α).
2. Let R ∈ {0, ∞}. Then λ1R (α) = λ1R (β) ⇔ ∃C ≥ 1: C1 α ≤ β ≤ Cα.
< ∞ ⇒ λpR (α) × λpR (α) ∼
α p
3. sup 2j+1αj = λR (α) for R ∈ {0, +∞} and p ∈
[1, ∞]. In particular, s × s ∼= s and s(Z) ∼
= s(N).
< ∞ ⇔ sup ln
1 p
P αj  j
4. λ∞ (α) = λ∞ (α) ∀p ∈ [1, ∞] ⇔ ∃r < 1: jr αj : j < ∞.

5. λ10 (α) = λp0 (α) ∀p ∈ [1, ∞] ⇔ ∀r < 1: < ∞ ⇔ limj→∞ ln j

P αj
jr αj = 0.

Proof. 1 ϕ : λpR (α) → λp0 (α), x 7→ (eRαj xj )j∈N is an isomorphism, since

erαj eRαj xj = e(R+r)αj xj for r < 0 (⇔ R + r < R).
1
2 (⇐) C1 α ≤ β ≤ Cα ⇒ k(e C r αj xj )j k`p ≤ k(er βj xj )j k`p ≤ k(eC r αj xj )j k`p and
{ C1 r : r < R} = {r : r < R} = {C r : r < R} for R = 0 and similarly for R = ∞.
(⇒) Let γj := max{αj , βj } for R = ∞, resp. γj := min{αj , βj } for R = 0. Then
λ1R (γ) ⊆ λ1R (α) ∩ λ1R (β) and the inclusion is continuous, since |xj erαj | ≤ |xj erγj |
for all 0 < r < R = +∞ resp. all r < R = 0. Moreover, λ1R (γ) = λ1R (α) ∩ λ1R (β),
since x ∈ λ1R (α) ∩ λ1R (β) ⇒ ∀r < R :
X X X X X
∞> |xj erαj | + |xj erβj | ≥ |xj erαj | + |xj erβj | ≥ |xj erγj |.
j j j,αj =γj j,βj =γj j

andreas.kriegl@univie.ac.at c July 1, 2016 13

1.26

By the open-mapping theorem λ1R (α) = λ1R (γ) = λ1R (β) as Fréchet spaces.
1.24 λ1 (α) λ1 (β)
⇒ ∀r ∃s > r ∃C > 0 : k kr R
====
= ≤ C k ks R
[Kri14, 1.3.3,1.3.7]
⇒ ∀r ∃s > r ∃C > 0 : Usβ ⊆ C Urα
===============
⇒ ∀y ∈ h(Urα )o ilin.sp : kyk∗s,β ≤ Ckyk∗r,α

⇒ ar ∈ (Urα )o ⊆ C (Usβ )o , since ∀x ∈ Urα : 1 ≥ kxkr := kar · xk`1

1.24 na o
j,r
⇒C ≥ kar k∗s,β =
==== = sup = er αj −s βj : j
bj,s
⇒ sup{r αj − s βj : j} ≤ ln C
  
 αj ≤ 1 ln C + s ≤ C 0 in case r > 0,
βj r βj
⇒ αj 
 ≥ 1 − ln C + (−s) ≥ C 0 > 0 in case r < 0.
βj −r βj

3 Let Φ : λpR (α) → λpR (α) × λpR (α) be given by

Φ(x) := (xeven , xodd ) := ((x2n )n∈N , (x2n+1 )n∈N ).
(r > 0) Then kxσ kr = kj 7→ eαj r x2j+σ k`p ≤ kj 7→ eα2j+σ r x2j+σ k`p ≤ kxkr and
kxkr = kj 7→ eαj r xj k`p = kj 7→ eα2j r x2j k`p + kj 7→ eα2j+1 r x2j+1 k`p
0 00
≤ kj 7→ eαj r x2j k`p + kj 7→ eαj r x2j+1 k`p = kxeven kr0 + kxodd kr00 ,
α2j α2j+1 α2j
where R > r0 > r sup αj and R > r00 > r sup αj > r sup αj .

1.25 P a
4 λ1∞ (α) = λp∞ (α) ⇐ ⇒ ∀r ∃s(> r) : j (er−s )αj = j aj,r
P
==== j,s
<∞⇔
r−s
P αj
⇔ ∃q = e < 1 : j q < ∞ ⇔ ∃δ > 0 : δ ln j ≤ αj :
(⇐) q αj ≤ e(r−s)δ ln j = j δ(r−s) ≤ j −2 , provided s > r + 2δ .
(⇒) ln j ln jn
αj unbounded ⇒ ∀n ∃jn : αjn ≥ n, w.l.o.g. jn+1 ≥ 2jn ≥ 8. Then for
−x
q = e with x > 0 we have
X X jn
X X
q αj = e−x αj ≥ (jn − jn−1 ) e−x αjn
n j=jn−1 +1 n
| {z }
j
≥jn /2
ln( j2n )− n
x
X X
ln(jn )
≥ e ≥ 1,
n n≥2x
x/n 1−x/n
since ln( j2n ) ≥ ln(jn ), or equivalently jn ≥ 41/2 = 2.
1.25 aj,r
5 λ10 (α) = λp0 (α) ⇐====⇒ ∀r < 0 ∃ (r <)s < 0 : r−s αj
P P
j (e ) = j aj,s < ∞ ⇔
∀q = er−s < 1 : j q αj < ∞ ⇔ limj→∞ ln j
P
αj = 0:
ln j
(⇐) limj→∞ αj = 0 ⇒ ∀x ∃N ∀j ≥ N : ln j x
αj < 2 ⇒
X X 2
X 1
e−x αj ≤ e−x x ln j = <∞
j2
j≥N j≥N j≥N

ln j ln jn
(⇒) limj→∞ αj 6= 0 ⇒ ∃δ > 0 ∀n ∃jn : αjn ≥ δ, w.l.o.g. jn+1 ≥ 2jn ≥ 8. Then
−x δ
for q := e with x := 2 we have
jn
jn
)− x
X X X X X X
q αj = e−x αj ≥ (jn − jn−1 ) e−x αjn ≥ eln( 2 δ ln(jn ) ≥ 1,
n j=jn−1 +1 n
| {z } n n
j
≥jn /2

14 andreas.kriegl@univie.ac.at c July 1, 2016

 1.26

1/2 1−1/2
since ln( j2n ) ≥ ln(jn ), or equivalently jn ≥ 41/2 = 2.

andreas.kriegl@univie.ac.at c July 1, 2016 15

1.26

16 andreas.kriegl@univie.ac.at c July 1, 2016

 2. Colimit closed (coreflective) subcategories

In this section we describe ((reduced) inductive) colimits of locally convex spaces.

And we consider the classes of (ultra-)bornological and (infra-)barrelled spaces, all
of which are invariant under the formation of colimits. We give descriptions of
Köthe sequence spaces as colimits of Banach spaces.

Barrelled and bornological spaces

2.1 Definition. Bornological and barrelled spaces.

An lcs is bornological (cf. [Kri14, 2.1.7], [MV92, 24.9 p.262], [Jar81, 13.1
p.272]) if bounded linear mappings (i.e. being bounded on bounded sets) on it
are continuous, or equivalently, every bornivorous (i.e. absorbing each bounded
set) absolutely convex subset is a 0-neighborhood (See [MV92, 24.10 p.263]).
An lcs is ultrabornological (See [Kri14, 5.4.20], [MV92, 24.14 p.264], [Jar81,
13.1 p.272]) if all linear maps on it, which are bounded on the Banach-disks (i.e.
absolutely convex bounded sets B for which EB is complete), are continuous, or
equivalently, every absolutely convex subset, which absorbs all Banach-disks, is a
0-neighborhood.
An lcs is called barrelled (german: tonnelliert) (See [Kri14, 4.2.1], [Jar81,
11.1 p.219], [MV92, Def. in 23.19 p.252]) if every barrel (german: Tonne) (i.e.
closed absolutely-convex absorbing subset) is a 0-neighborhood, equivalently, the
uniform boundedness theorem holds (cf. [Kri14, 4.2.2]).
An lcs is called infra-barrelled (german: quasi-tonneliert, infra-tonne-
liert ) (See [Kri14, 5.4.20], [Jar81, 11.1 p.219], [MV92, Def. in 23.19 p.252]) if
every bornivorous barrel is a 0-neighborhood, equivalently, E embeds topolog-
ically into the bidual (See [Kri14, 5.4.20]).
Since obviously “bornivorous⇒absorbs Banach disks” and barrels absorb Banach-
disks by the Banach-Mackey-Theorem (See [Kri07b, 7.4.18]) we have the following
implications (See [MV92, 24.12 p.263], [MV92, 24.15 p.264]):
bornivorous barrel infra-barrelled
19 fn
px /'
barrel bornivorous barrelledem bornological
08
.& ow
absorbs Banach-disks ultrabornological
For sequentially complete or at least locally-complete lcs (i.e. the Banach-disks
form a basis for the bornology) the implications from left to right can clearly be
inverted (See [MV92, 23.20 p.252],[MV92, 23.21 p.253]).

2.2 Lemma. (See [Kri14, 2.1.6], [Jar81, 10.1.4 p.197]).

In any metrizable lcs every convergent sequence is Mackey-convergent.

andreas.kriegl@univie.ac.at c July 1, 2016 17

2.4 Barrelled and bornological spaces

A sequence (xn )n∈N in an lcs is called Mackey convergent towards x∞ iff there
exists a sequence λn → ∞ in R with {λn (xn − x∞ ) : n ∈ N} being bounded.
(k)
Proof. Let (pk )k∈N be a basis of seminorms. Since for each k the sequence µn :=
pk (xn − x∞ ) → 0 for n → ∞ we find another sequence 0 6= µ∞ k ∞
n → 0 with {µn /µn :
n ∈ N} bounded for each k (See [Kri14, 2.1.6]). Then λn := 1/µk has the required
∞

property.

2.3 Corollary (See [MV92, 23.23 p.253], [Kri14, 4.1.11], [Kri14, 4.2.4] ).
Fréchet spaces are ultrabornological, hence bornological, barrelled and infrabarrelled.

Proof. Metrizable lcs are bornological (See [Kri14, 2.1.7], [MV92, 24.13 p.264]),
since any bounded linear mapping f on them is (sequentially) continuous: Let xn →
x∞ , then by 2.2 there are λn → ∞ with n 7→ f (λn (xn −x∞ )) = λn (f (xn )−f (x∞ ))
bounded, hence f (xn ) → f (x∞ ). Completeness implies now that the space is even
ultrabornological.

2.4 Colimits.
Let F : J → lcs be a functor from a partially ordered set (J, )op = (J, ≺) or even
from a small category J into that of locally convex spaces. The colimit colim F
of F (See [Kri08, 3.25]) is then given as quotient of the coproduct (direct sum,
cf. [Kri14, 3.6.1])
a n Y o
F(j) := x ∈ F(j) : xj = 0 for all but finitely many j
j j
`
with the final locally convex structure with respect to the inclusions Ej ,→ j F(j)
(whose continuous seminorms are those which restricted to each summand F(j)
are seminorms of F(j)), where we factor out the congruence relation generated
0
x(j) ∼ (F(f )(x))(j ) for every j ≺ j 0 (morphism f : j → j 0 in J ), where x(j)
denotes the point with j-th coordinate x ∈ F(j) and all other coordinates equal to
0. Since the topology on this quotient need not be Hausdorff, one has to factor out
the closure of {0} in addition, i.e. the intersection of the kernels of all its seminorms.
In the particular case, where J op F = (J, ) is directed, the first (not necessarily
Hausdorff) quotient is given by j F(j)/ ∼, where x1 ∈ F(j1 ) is equivalent to
x2 ∈ F(j2 ) iff for some j  j1 , j2 : F(j1 ≺ j)(x1 ) = F(j2 ≺ j)(x2 ). In this case the
colimit is also called inductive limit (See [Jar81, 4.5 p.82]) and denoted lim F.
−→
An inductive limits is called reduced, iff all ιj : F(j) → lim F are injective.
−→
By replacing F(j) with the image F̃(j) of ιj in lim F supplied with its quotient
−→
structure, we get that lim F equals lim F̃, which is a reduced inductive limit (See
−→ −→
[Jar81, 4.5.2 p.82]). Note for this that F̃(j ≺ j 0 ) is then a well defined injective
continuous linear mapping.
An even more restricted situation is, when J = (N, ≤), i.e. we have an inductive
limit of a sequence of spaces (the steps of the limit). The inductive limit of a
sequence of Fréchet-spaces (a so-called (LF)-space) is almost never a Fréchet space
(See [Kri14, 4.1.13]): Strict inductive limits of sequences (i.e. En is a closed
topological subspace in En+1 for each n), which are not finally constant, can not
be Baire spaces and hence are not Fréchet; And, more generally, by [Jar81, 12.4.4
p.259] a metrizable space with a countable base of bornology has to be normed, in
particular this is valid for (locally) complete (LB)-spaces (See [Flo73, 5.5 p.73]),
i.e. inductive limits of a sequenceSof Banach spaces. Even more generally, if all Fn
and F∞ := limn Fn (hence F∞ = n∈N ιn (Fn )) are Fréchet, then by Grothendieck’s
−→

18 andreas.kriegl@univie.ac.at c July 1, 2016

Barrelled and bornological spaces 2.6

factorization theorem 2.6 F∞ ⊆ ιn (Fn ) for some n.

Furthermore, it is not true in general that (LB)-spaces are complete and regular
(See [Mak63, Beispiel 2]), i.e. bounded sets are contained and bounded in some
step, or, stronger, converging sequences (resp. compact subsets) are converging
(resp. compact) in some step.

2.5 Stability under colimits.

Colimits of bornological spaces Ej are again bornological (See [Jar81, 13.1.5 p.273],
[MV92, 24.16 p.264]), since bounded linear mappings on colimj Ej are bounded
mappings on each Ej and hence continuous on Ej , and by the universal property
of the limit also continuous on colimj Ej .
By definition any bornological space E is the inductive limit of the spaces EB ,
where B runs through the bounded (closed) absolutely convex subsets. Thus the
bornological spaces are exactly the colimits of normed spaces.
The same argument works for ultrabornological instead of bornological, since the
continuous images f (B) of Banach disks B are again Banach disks: EB → Ff (B) is
a quotient mapping, since

pf (B) (f (x)) = inf λ > 0 : f (x) ∈ λ · f (B) = f (λ B), i.e. ∃b ∈ B : f (x − λ b) = 0

= inf λ > 0 : ∃b ∈ B ∃z ∈ ker f : λ b − x = z

= inf λ > 0 : ∃z ∈ ker f ∃b ∈ B : x + z = λ b
n  o
= inf inf λ > 0 : ∃b ∈ B : x + z = λ b : z ∈ ker f

= inf pB (x + z) : z ∈ ker f = pf B (f (x))

is the quotient norm (See [Kri14, 4.3.6]).

Furthermore, (infra-)barrelled spaces are stable under colimits (See [Jar81, 11.3.1.c
p.223]): For quotients this follows since inverse images of barrels are barrels and of
bornivorous sets are bornivorous. For coproducts it can be found in [Jar81, 8.8.10
p.168]

2.6 Grothendiecks factorization theorem (See [MV92, 24.33 p.271]).

Let F be an lcs, let E and En for n ∈ N be FréchetS spaces, fn ∈ L(En , F ) and
f ∈ L(E, F ) continuous linear mappings. If f (E) ⊆ n∈N fn (En ) then there exists
an m ∈ N with f (E) ⊆ fm (Em ). If, in addition, fm is injective, then there exists
an f˜ ∈ L(E, Em ) with f = fm ◦ f˜.

E
f
/ S fn (En )   /F
n O

f˜
 % ?
∃Em / fm (Em )
∃fm

Proof. Let Gn := {(x, y) ∈ E × En : f (x) = fn (y)} = graph(fn−1 ◦ f ) be the

pull-back of f and fn , a closed linear subspace of the Fréchet space E
−1
S × En . Then
pr1 (G n ) = {x ∈ E : ∃y : f (x) = fn (y)} = f (f n (E n )) and hence n pr1 (Gn ) =
f −1 ( n fn (En )) = E. By the theorem of Baire (see [Kri14, 4.1.11]), there exists
S
an m such that pr1 (Gm ) is not meagre, hence by the open mapping theorem (see
[Kri14, 4.3.6]) pr1 : Gm → E is onto, i.e. f (E) = f (f −1 (fm (Em ))) ⊆ fm (Em ).

andreas.kriegl@univie.ac.at c July 1, 2016 19

2.10 Barrelled and bornological spaces

If, in addition, fm is injective, then f˜ := fm

−1
◦ f : E → Em is a well-defined linear
mapping with closed graph Gm , hence is continuous by the closed graph theorem
(see [Kri14, 4.3.1]).

2.7 Lemma (See [MV92, 24.34 p.272]).

Let an lcs E carry the final structure with respect to countable
S many continuous
linear mappings fn : En → E for Fréchet spaces En with n fn (En ) = E.
Then E is the (reduced) inductive limit of a sequence of Fréchet spaces.

Proof. We construct a strictly increasing sequence (nk )k in N with j≤nk fj (Ej ) ⊆

S
`
fnk+1 (Enk+1 ). For the Fréchet space F := j≤nk Ej consider the continuous linear
P
map f : (xj )j≤nk 7→ j≤nk fj (xj ), F → E. By 2.6 there exists an nk+1 such that
j≤nk fj (Ej ) = f (F ) ⊆ fnk+1 (Enk+1 ). Let Ẽk := Enk / ker fnk 
S
→ fnk (Enk ) ,→ E.
The mapping Ej → Ẽk+1 for j ≤ nk has closed graph, hence is continuous by the
closed graph theorem, and thus also Ẽk → Ẽk+1 . The inductive limit structure on
E of the increasing sequence of Fréchet spaces Ẽk is finer than the given one since
Ẽk  E is continuous. Because of fj (Ej ) ⊆ Ẽnk for j ≤ nk it is also coarser.

2.8 Corollary. All representations of an (LF) space are equivalent

(See [MV92, 24.35 p.273]).
(i)
Let E be the reduced inductive limit of two sequences of Fréchet spaces (En )n∈N
for i ∈ {0, 1}. Then ∀n ∈ N ∃k ∈ N : En embeds continuously into Ek (and
(0) (1)

(1) (0)
similarly En into Ek ).

2.9 Elements in λ∞ (A) (See [MV92, 27.4 p.308]).

1. b ∈ λ∞ (A) ⇔ ∀a ∈ A ∃Ca > 0: |bj | ≤ inf a∈A Ca /aj .

2. If A is countable, then ∃b ∈ λ∞ (A) ∀j : bj > 0.
3. If A is countable, then ∀b ∈ λ∞ (A) ∃b0 ∈ λ∞ (A) ∀j : 0 6= b0j ≥ |bj |.

Proof. ( 1 ) b ∈ λ∞ (A) ⇔ ∀a: b · a bounded (by Ca > 0), i.e. ∀j: |bj | ≤ Ca /aj ⇔
∀j: |bj | ≤ inf a Ca /aj .
( 2 ) Let A := {a(k) : k ∈ N} with k 7→ a(k) increasing. For each k ∈ N choose
(k) (k) (k) (k)
Ck > k max{1, a0 , . . . , ak }. ⇒ Ck /aj ≥ k for all k ≥ j. ⇒ bj := inf k Ck /aj =
(k)
mink Ck /aj > 0 and b ∈ λ∞ (A).
( 3 ) By ( 2 ) there is a b0 ∈ λ∞ (A) with b0j > 0 forall j. For b ∈ λ∞ (A) also
b00 : j 7→ max{|bj |, b0j } is in λ∞ (A) and satisfies b00 ≥ |b| and ∀j : b00j ≥ b0j > 0.

2.10 Bounded sets in λp (A) (See [MV92, 27.5,27.6 p.309]).

For 1 ≤ p ≤ ∞ the sets Bbp := {x : kx/bk`p ≤ 1} for b ∈ λ∞ (A) form a basis of the
bornology of λp (A) if p = ∞ or if A is countable.
The sets Bbo := Bb∞ ∩ c0 (A) for b ∈ λ∞ (A) form a basis of the bornology of c0 (A).

Proof. b ∈ λ∞ (A) ⇒ Bbp ⊆ λp (A) bounded, since x ∈ Bbp ⇒ carr x ⊆ carr b and
∀a ∈ A: kxka = kx · ak`p = kx · 1b · b · ak`p ≤ kx/bk`p · kb · ak∞ ≤ 1 · kb · ak∞ .
Conversely, let B ⊆ λ∞ (A) be bounded, i.e. ∀a ∈ A ∃Ca > 0 ∀x ∈ B: kx·ak`∞ ≤ Ca .
Let bj := inf{ Caja : a ∈ A}, which is < ∞, since aj > 0 for some a. Then b ∈ λ∞ (A),
since |bj aj | ≤ Ca for all a ∈ A and j ∈ N. Furthermore, since |xj Cja | ≤ 1 for all
a

a ∈ A and j ∈ N, we get |xj bj | = supa∈A |xj Ca | ≤ 1, i.e. kx · b k` ≤ 1 for all

1 aj 1 ∞

20 andreas.kriegl@univie.ac.at c July 1, 2016

Barrelled and bornological spaces 2.12

x ∈ B.
Since c0 (A) is a subspace of λ∞ (A), this works for c0 (A) as well.
Now for 1 ≤ p < ∞ and A = {a(k) : k ∈ N} countable: Let B ⊆ λp (A) be bounded,
2.9.1
⇒ b := inf k 2k+1 Ck /a(k) ∈ λ∞ (A).
i.e. ∀k ∃Ck > 0 ∀x ∈ B : kxkk ≤ Ck =====
(k) (k)
1 aj X aj
= sup k+1 ≤ ⇒
bj k 2 Ck 2k+1 Ck
k
X xa(k) X kxa(k) k`p X kxkk X 1
kx/bk`p ≤ ≤ ≤ ≤ =1
2k+1 Ck `p 2k+1 C k 2k+1 Ck 2k+1
k k k k
⇒x∈ Bbp , i.e. B ⊆ Bbp .

2.11 Counter-example.
Let A := `p for 1 ≤ p < ∞. Then λ∞ (A) := {x ∈ KN : ∀y ∈ `p : kx · yk`∞ < ∞} is
the linear space `∞ :
(⊇) x ∈ `∞ , y ∈ `p ⊆ `∞ ⇒ kx · yk`∞ ≤ kxk`∞ · kyk`∞ .
(⊆) Suppose x ∈ λ∞ (A) is unbounded ⇒ ∃jn (W.l.o.g. strictly increasing) with
|xjn | ≥ n 2n . (
2−n for j = jn
yj := .
0 otherwise
/ λ∞ (A).
Then kyk`p ≤ kyk`1 = n 21n < ∞, but kx · yk`∞ ≥ |xjn yjn | ≥ n, i.e. x ∈
P

Note that λr (A) = `∞ as linear spaces for all p ≤ r ≤ ∞:

`∞ ⊆ λp (A) ⊆ λr (A) ⊆ λ∞ (A) = `∞ , since kx · yk`p ≤ kxk`∞ · kyk`p .
Now let s0 := 0 and recursively sn+1 := sn + n and put
(
(n) 1 for sn ≤ j < sn+1
xj :=
0 otherwise
and B := {x(n) : n ∈ N}. Then B is bounded in λp (A), since
X−1
sn+1 1/p
(n)
kx · yk`p = |yj |p ≤ kyk`p .
j=sn

However, there is no b ∈ λ∞ (A) = `∞ such that kx · 1b k`p ≤ 1 for all x ∈ B:

In fact, let β := kbk`∞ < ∞ then
1 1 n1/p
x(n) · ≥ kx(n) k`p = → ∞.
b `p β β

2.12 λp (A) as colimit of (uncountable many) `p ’s for countable A

(See [MV92, 27.7 p.309]).
There exists a basis B for the bornology of λp (A) with λp (A)B ∼
= `p for all B ∈ B.

∀b ∈ λ∞ (A) : `p (carr b) −∼→ hBbp i = (λp (A))Bbp ⊆ λp (A)

·b
=

Proof. 2.9.3 ⇒ ∀b ∈ λ∞ (A) ∃b0 ∈ λ∞ (A) ∀j: 0 6= b0j ≥ |bj | ⇒ Bbp ⊆ Bbp0 and
λp (A)B p0 ∼
= `p .
b

andreas.kriegl@univie.ac.at c July 1, 2016 21

2.12 Barrelled and bornological spaces

22 andreas.kriegl@univie.ac.at c July 1, 2016

 3. Limit closed (reflective) subcategories

In the following sections we consider classes of locally convex spaces which are
invariant under the formation of limits, i.e. various completeness conditions, semi-
reflexivity, Montel spaces, Schwartz spaces, and nuclear spaces. And we characterize
those Köthe sequence spaces having these properties.

Completeness, compactness and (DN)

In this section we consider various completness conditions. And we discuss (pre-)com-

pact subsets and operators, since they are relevant for the classes to follow. We
introduce the property (DN) which allows to differentiate between power series
spaces of finite and of infinite type.

3.1 Completeness.
For lcs E we consider the following completeness conditions:
• E is called complete iff evevry Cauchy net (or Cauchy filter) converges.
• E is called quasi complete iff every closed bounded subset is complete.
• E is called sequentially complete iff Cauchy sequences converge.
• E is called locally complete (or Mackey-complete) iff EB is a Banach
space for every closed absolutely convex bounded subset B ⊆ E.
One obviously has the implications:
complete ⇒ quasi-complete ⇒ sequentially complete ⇒ locally complete.
For metrizable spaces all 4 conditions are equivalent (See [Kri14, 2.2.2]). Each
of these completeness properties is inherited by closed subspaces ([Kri14, 3.1.4]),
products ([Kri14, 3.2.1]), and coproducts ([Kri14, 3.6.1]) (See [Jar81, 3.2.5 p.59],
[Jar81, 3.2.6 p.59], [Jar81, 6.6.7 p.111]).
The completion (i.e. reflector) of any lcs E is given by the space of all linear func-
tionals on E ∗ , whose restrictions to equicontinuous subsets are σ(E ∗ , E)-continuous,
supplied with the topology of uniform convergence on the equicontinuous subsets,
see [Kri14, 5.5.7].

3.2 Precompact sets.

A subset K ⊆ E in an lcs is called precompact iff
[
∀U ∃F ⊆ E finite : K ⊆ F + U = (y + U ).
y∈F

This is exactly the case, when K is relatively compact in the completion of E

(See [Kri07a, 6.2]). The precompact subsets of a product of lcs’s are those whose
projections to the factors are precompact; The precompact subsets of a coproduct
of lcs’s are those whose projections to the summands are precompact and are almost
always {0} (See [Kri07a, 6.3]).

andreas.kriegl@univie.ac.at c July 1, 2016 23

3.8 Completeness, compactness and (DN)

3.3 Mackey-Arens Theorem

(See [Kri14, 5.4.15], [Jar81, 8.5.5 p.158], [MV92, 23.8 p.247]).
The finest topology compatible with a dual pairing (E, F ) is the Mackey-topology
µ(E, F ), i.e. the topology of uniform convergence on σ(F, E)-compact absolutely
convex subsets of F .

3.4 Alaŏglu-Bourbaki Theorem

(See [Kri14, 5.4.12], [Jar81, 8.5.2 p.157], [MV92, 23.5 p.245]).
Each equicontinuous set is relatively compact with respect to τpc (E ∗ , E), the topol-
ogy of uniform convergence on precompact subsets, or equivalently, with respect to
σ(E ∗ , E).

Proof of the equivalence (See [Jar81, 8.5.1.b p.156]). Let U ⊆ E be a 0-

neighborhood, x∗ ∈ U o , and A ⊆ E be precompact, i.e. x∗ + Ao a typical neigh-
borhood of x∗ with respect to τpc (E ∗ , E). Thus there is a finite set F ⊆ E with
3A ⊆ F +U , hence (x∗ +F o )∩U o ⊆ x∗ +Ao , since for all y ∗ ∈ U 0 with y ∗ −x∗ ∈ F o
and a ∈ A exist y ∈ F and u ∈ U with 3a = y + u and hence
1 1 ∗ 
(y ∗ − x∗ )(a) = (y ∗ − x∗ )(y + u) ≤ (y − x∗ )(y) + y ∗ (u) + x∗ (u)
3 3
1
≤ (1 + 1 + 1) = 1, i.e. y ∗ − x∗ ∈ Ao .
3

3.5 Proposition (See [Jar81, 8.5.3 p.157]).

E separable ⇒ equicontinuous subsets are σ(E ∗ , E)-metrizable.

Proof. Let D := {xj : j ∈ N} ⊆ E be dense and let E0 be the linear span of

D.
P∞Then σ(E ∗ , E0 ) is Hausdorff. Let W be a 0-nbhd. for σ(E ∗ , E0 ), i.e. ∃yi =
i i o
j=1 λj xjP∈ E0 with λj = 0 for almost all j, say j ≤ m, and {y1 , . . . , yn } ⊆ W .
For max{ j |λj | : i} < λ ∈ Q we have λ · {x1 , . . . , xm } ⊆ {y1 , . . . , yn } ⊆ W .
i 1 o o

Thus σ(E ∗ , E0 ) is metrizable and coincides with σ(E ∗ , E) on equicontinuous sets:

In fact, E0 ⊆ E ⇒ σ(E ∗ , E) → σ(E ∗ , E0 ) is continuous. Conversely, let U be a
0-nbhd in E, x∗ ∈ U o , ε > 0, and xi ∈ E. Choose x̃i ∈ E0 with x̃i − xi ∈ 3ε U . For
y ∗ ∈ U o ∩ (x∗ + 3ε {x̃1 , . . . , x̃k }o ) we have:
ε
|(y ∗ − x∗ )(xi )| ≤ |y ∗ (xi − x̃i )| + |x∗ (xi − x̃i )| + |(y ∗ − x∗ )(x̃i )| ≤ 3 ,
3
i.e. y ∗ ∈ U o ∩ (x∗ + ε{x1 , . . . , xk }o ).

3.6 Lemma. Compact subsets of Fréchet spaces

(See [Kri07b, 6.4.3 p.119], [Jar81, 10.1.1 p.196]).
A subset of a Fréchet space is precompact (equivalently, relatively compact) if and
only if it is contained in the closed convex hull of some 0-sequence.

3.7 Definition. Compact operator.

A linear operator between Banach spaces is called (weakly) compact if the image
of the unit ball is (weakly) relatively compact.
A linear operator between Hilbert spaces is compact iff it can be approximated by
finite dimensional operators with respect to the operator norm, see [Kri07b, 6.4.8].

3.8 Lemma. Orthogonal representation of compact operators

(See [Kri07a, 5.3], [Jar81, 20.1.2 p.452]).
An operator T between Hilbert spaces is P
compact iff there are orthonormal sequences
en and fn and λn → 0 such that T x = n λn hen , xifn .

24 andreas.kriegl@univie.ac.at c July 1, 2016

Completeness, compactness and (DN) 3.10

Proof. (⇐) If T has such a representation, then the finite sums define finite
dimensional operators which converge to T .
(⇒) Since any compact T : E → F induces a compact injective operator T :
(ker T )⊥ → T (E) with dense image, we may assume that T is injective. Now we
consider the positive compact operator T ∗ T . Its eigenvalues are all non-zero, since
T ∗ T x = 0 implies kT xk2 = hT x, T xi = hT ∗ T x, xi = 0. By [Kri07b, 6.5.4] there is
an orthonormal P sequences of Eigen-vectors en with Eigen-value 0 6= λ2n → 0 such
∗
that T T x = n λn hen , xien . Let fn := λ1n T en . Then a simple direct calculation
2
P
shows that the fn are orthonormal. Note that x = n hen , xien . Otherwise the
compact positive operator T ∗ T restricted to the orthogonal complement {ek : k}⊥
would have a unit Eigen-vector e with positive P Eigen-value λ. Which is impossible
by definition of the ek . So we obtain T x = n hen , xiλn fn .
Another way to prove this is to use the polar decomposition T = U |T |, see [Kri14,
7.24], where U is a partial isometry and |T | a positive and also compact operator.
The spectral P theorem for |T | gives an orthonormal family en and λ ∈ c0 , such
that T x = k λk hek , xiek . Applying U to this equation, shows that we may take
fk := U ek .

3.9 Corollary (See [Kri07a, 5.4], [Jar81, 20.1.3 p.453]).

An operator T between Hilbert spaces is compact iff hT en , fn i → 0 holds for all
orthonormal sequences en and fn .

Proof. (⇒) Since |hT en , fn i ≤ kT en k · kfn k = kT en k it is enough to show that

T en → 0. Since en converges weakly to 0 (in fact hx, en i is even quadratic sum-
mable) we conclude that T en converges to 0 weakly. Since en is contained in the
unit-ball and T is compact, every subsequence of T en has a subsequence, which is
convergent. And the limit has to be 0, since this is true for the weak topology. But
from this it easily follows that T en → 0.
(⇐) Given ε > 0 we choose maximal orthonormal sequences (ei )i∈I and (fi )i∈I such
that |hT ei , fi i| ≥ ε.
PBy assumption I mustPbe finite. We consider the orthonormal
projections P := i∈I ei ⊗ ei and Q := i∈I fi ⊗ fi . For the composition with
the ortho-projections on the complement we obtain (1 − Q)T (1 − P ) = T − (T P +
QT − QT P ) =: T − S. Hence S is a finite dimensional operator and we claim that
kT −Sk ≤ ε. Suppose this were not true. Then there is an x with k(T −S)xk > ε kxk
and hence an y such that |hT (1 − P )x, (1 − Q)yi| = |h(T − S)x, yi| > ε kxk kyk. Let
e0 := (1 − P )x and f0 := (1 − Q)y. Obviously e0 , f0 6= 0 and hence we may assume
without loss of generality that ke0 k = 1 = kf0 k and hence kxk ≥ 1 and kyk ≥ 1.
Since e0 ∈ (1−P )(E) ⊆ P (E)⊥ = {ei : i ∈ I}⊥ and f0 ∈ (1−Q)(F ) ⊆ {fi : i ∈ I}⊥
we get a contradiction to the maximality of I.

3.10 Compact diagonal operators between `p ’s (See [MV92, 27.8 p.309]).

Let ` := `p with 1 ≤ p < ∞ or c0 ⊆ ` ⊆ `∞ invariant under multiplication with `∞ .
Let D : ` → ` be a diagonal-operator with coeffcients d ∈ `∞ .
(1) D is compact
⇔ (2) d ∈ c0
⇔ (3) D is weakly-compact in case ` = `1 .

Proof. ( 1 ⇒ 2 ) Let
(
xj /dj for |dj | ≥ ε,
Tε : ` → `, Tε (x)j :=
0 elsewhere.

andreas.kriegl@univie.ac.at c July 1, 2016 25

3.12 Completeness, compactness and (DN)

⇒ Pε := D ◦ Tε is a compact projection ⇒ Pε (`) = ker(1 − Pε ) = {x ∈ ` : carr x ⊆

{j : |dj | ≥ ε}} is finite dimensional (by [Kri14, 3.4.5]) ⇒ {j : |dj | ≥ ε} is finite.
( 2 ⇒ 1 ) Pn : x 7→ x · χ{1,...,n} ⇒ kD − D ◦ Pn k ≤ sup{|dj | : j > n} → 0, since
d ∈ c0 , i.e. D is compact as limit of fin.dim. operators.
( 1 ⇒ 3 ) is trivial

( 3 ⇒ 2 ) Pε := D ◦ Tε is weakly-compact. Suppose {j : |dj | ≥ ε} infinite ⇒

Pε (`) ∼
= `1 and the closed unit disk in `1 is weakly compact ⇒ `1 reflexive (see
3.17 ), a contradiction.

3.11 Approximation numbers for diagonal operators on `2 .

Let D : `2 → `2 be a diagonal-operator with coefficients d ∈ `∞ with |di | & 0. Then
its approximation numbers are
n o
an (D) := inf kD − T k : dim T (`2 ) ≤ n = dn .

Proof (See [MV92, Aufgabe 16.(3) p.392]). Note that kDk = kdk`∞ = sup{|di | :
i ∈ N} since kD(x)k = kd · xk`2 ≤ kdk`∞ · kxk`2 and D(e(k) ) = dk e(k) .
Thus an (D) ≤ kDn k = sup{|dk | : k ≥ n}, where Dn is the diagonal operator with
entries d · χ[n,∞) with dim((D − Dn )(`2 )) = n. Conversely, let dim T (`2 ) ≤ n. Then
Pn
∃y = i=0 yi ei with kyk`2 = 1 and T (y) = 0. Thus kD − T k`2 ≥ k(D − T )yk`2 =
Pn
kDyk`2 = ( i=0 |di yi |2 )1/2 ≥ min{|di | : i ≤ n} kyk`2 = |dn |.

3.12 Proposition. Equality λr (α) = λr (β) (See [MV92, 29.1 p.338]).

For r ∈ {0, +∞} let λr := λ2r .

(1) λr (α) ∼
= λr (β);
⇔ (2) λr (α) = λr (β) as lcs;
⇔ (3) λr (α) = λr (β) as sets;
⇔ (4) ∃C > 0 ∃n0 ∈ N ∀n ≥ n0 : 1
C αn ≤ βn ≤ Cαn .

Proof. (4 ⇒ 3) is obvious.
(3 ⇒ 2) apply the closed graph theorem using that convergence in λr implies
coordinatewise convergence.
(2 ⇒ 1) is obvious.
(1 ⇒ 4) Let Λα
s := λr (α)s := λr (α)/ ker k ks for s < r.

A : λr (α) → λr (β) iso, B := A−1 ⇒

⇒∀t < r ∃s < r ∃C > 0 : kAxkt ≤ Ckxks ,
∀s < s0 < r ∃t < t0 < r ∃D > 0 : kByks0 ≤ Dkykt0
β β 0 0 β β
⇒∃Ã ∈ L(Λα α t s α α
s , Λt ), B̃ ∈ L(Λt0 , Λs0 ) : ιt = Ã ◦ ιs ◦ B̃ : Λt0 → Λs0 → Λs → Λt
0 0 0 0 0
⇒ιss compact by 3.10 and an (ιtt ) = e(t−t )βn , an (ιss ) = e(s−s )αn by 3.11 .
0 0
Obviously an (ιtt ) ≤ kÃkan (ιss )kB̃k cf. 4.168
s0 − s log(kÃkkB̃k)
⇒βn ≤ Cαn + D for C := , D :=
t0 − t t0 − t

26 andreas.kriegl@univie.ac.at c July 1, 2016

Completeness, compactness and (DN) 3.17

λr (β)
B / λr (α) A / λr (β)

}} {{ ιss
0 ## !!
Λβt0
B̃ / Λα0 / Λβs à / Λβt
s 8

0
ιtt

3.13 Definition. Dominating norm (DN).

Let k kk be a monotone increasing basis of seminorms for the Fréchet space E.
Then E is said to have property (DN) iff
∃q ∀p ∃p0 ∃C ∀x : kxk2p ≤ Ckxkq kxkp0
It follows that k kq is a norm, a so-called dominating norm.

3.14 Inheritance properties of (DN) (See [MV92, 29.2 p.339]).

1. (DN) is topological invariant.

2. (DN) is inherited by closed subspaces.
3. λ∞ (α) has (DN).

Proof. ( 1 ) and ( 2 ) are obvious.

t2 −t0 t2 −t0 t0 α k
( 3 ) For t0 < t1 < t2 let p := t2 −t1 , q := t1 −t0 , fk := (e |xk |)2/p and gk :=
Hölder 2/p 2/q
(et2 αk |xk |)2/q = ⇒ (kxkt1 )2 = kf gk1 ≤
===== kxkt0 kxkt2 ⇒ (kxkk )2 ≤ kxk0 kxk2k .

3.15 Corollary (See [MV92, 29.3 p.339]).

λ0 (α) ∼
6 λ∞ (β) for all α, β % ∞.
=

Proof. By 3.14 λ∞ (β) has (DN). Indirectly, suppose λ0 (α) has (DN), i.e.
∃τ < 0 ∀t < 0 ∃T < 0 ∃C > 0 : kxk2t ≤ Ckxkτ kxkT .
1
x := ej ⇒ e2tαj ≤ C eτ αj +T αj ≤ C eτ αj ⇒ 2t ≤ αj ln(C) + τ , limj αj = +∞ ⇒
t ≤ τ , a contradiction.

Reflexive spaces

3.16 Definition. Reflexive spaces.

An lcs is called semi-reflexive iff the canonical mapping δ : E → E ∗∗ is onto.
An lcs is called reflexive iff the canonical mapping δ : E → E ∗∗ is an isomorphism
of lcs (See [MV92, Def. nach 23.17 p.251], [Kri14, 5.4.21], [Jar81, 11.4 p.227]).
Reflexive spaces are stable under products, coproduct and regular reduced inductive
limits. Semi-reflexive space are in addition stable under closed subspaces (See
[Jar81, 11.4.5 p.228]).

3.17 Characterizing semi-reflexivity

(See [MV92, 23.18 p.251], [Kri14, 5.4.22], [Jar81, 11.4.1 p.227]).
An lcs is semi-reflexive iff every bounded subset is relatively weakly compact.

andreas.kriegl@univie.ac.at c July 1, 2016 27

3.24 Reflexive spaces

Corollary (See [Jar81, 11.4.6 p.229]).

Semi-reflexive spaces are quasi-complete.

Proof. Let (xi ) be a Cauchy-net in a closed bounded B ⊆ E. Then (xi ) is Cauchy

for the weak topology and since B is weakly compact (xi ) converges weakly to some
x∞ . Let U be a closed absolutely convex 0-neighborhood. Thus xi − xi0 ∈ U finally,
and since U is also weakly-closed ([Kri14, 5.4.8]) xi − x∞ ∈ U finally.

3.18 Characterizing reflexivity

(See [MV92, 23.22 p.253], [Kri14, 5.4.23]), [Jar81, 11.4.2 p.228].
An lcs is reflexive iff it is semi-reflexive and (infra-)barrelled.

3.19 Corollary. Characterizing reflexive Fréchet spaces

(See [MV92, 23.24 p.253]).
A Fréchet space is reflexive iff every bounded subset is weakly relatively compact.

Proof. Since every (F) space is (infra-)barrelled by 2.3 the result follows from
3.18 .

3.20 λp (A) ist reflexive for 1 < p < ∞ (See [MV92, 27.3 p.307]).

1.13 , 3.16
Proof. `p reflexive = ⇒ λp reflexive.
=========

Montel spaces

3.21 Definition. Montel spaces.

An lcs is called semi-Montel space (See [MV92, Def. in 24.23 p.267], [Kri07a,
4.47,4.48 p.104]) iff all its bounded subsets are relatively compact.
An lcs is called Montel space (denoted (M) for short) (See [MV92, Def. in 24.23
p.267], [Kri07a, 4.47,4.48 p.104]) iff it is semi-Montel and infra-barrelled.

3.22 Montel spaces are reflexiv

(See [MV92, 24.24 p.267], [Jar81, 11.5.1 p.230]).
(Semi-)Montel spaces E are (semi-)reflexive and their σ(E, E ∗ )-convergent sequences
are convergent.

Proof. By definition bounded sets in semi-Montel spaces E are relatively compact

hence also relatively compact for the weak topology. Thus E is semi-reflexive by
3.17 . Since Montel spaces are infra-barrelled by definition, they are reflexive and
barrelled by 3.18 . Weakly convergent sequences are bounded, hence are relatively
compact for semi-Montel spaces, so the weak topology coincides with the given one
on this closure.

3.23 Inheritance properties of Montel spaces (See [Jar81, 11.5.4 p.230]).

Obviously closed subspaces, products and coproducts of semi-Montel spaces are
semi-Montel. Since barrelledness is inherited by products and coproducts (see 2.5 )
the same is true for the Montel property. The only normable (Semi-)Montel spaces
are the finite dimensional ones, see [Kri14, 3.4.5].

3.24 Proposition (See [Jar81, 9.3.7 p.179]).

γ(E ∗ , E) = τc (E ∗ , Ẽ), where γ(E ∗ , E) is the finest locally convex topology on

28 andreas.kriegl@univie.ac.at c July 1, 2016

Montel spaces 3.27

E ∗ = Ẽ ∗ into which all polars U o for 0-nbhds U in E (or the completion Ẽ) with
their compact topology continuously embed and τc (E ∗ , Ẽ) is the topology of uniform
convergence on compact subsets of the completion Ẽ.

Proof. Since for 0-nbhds U in E (or Ẽ) the polar U o is σ(E ∗ , Ẽ) compact and
even τpc (E ∗ , Ẽ) = τc (E ∗ , Ẽ) compact by 3.4 , we have γ ≥ τc (E ∗ , Ẽ) ≥ σ(E ∗ , Ẽ).
By Grothendieck’s completion result (See [Kri14, 5.5.7]) Ẽ = (E ∗ , γ)∗ , hence γ
is compatible with the duality (E ∗ , Ẽ), i.e. coincides with the topology of uniform
convergence on the closed equicontinuous subsets in (E ∗ , γ)∗ = Ẽ (see [Kri14,
5.4.11]). Let C be set of these subsets. All of them are compact for τpc (Ẽ, (E ∗ , γ))
by 3.4 . The identity (Ẽ, τpc (Ẽ, (E ∗ , γ))) → Ẽ is continuous, since Ẽ carries
the topology of uniform convergence on the equicontinuous subsets (polars U o ) in
Ẽ ∗ = E ∗ and polars U o are γ-compact by definition. Thus the sets in C are compact
in Ẽ. Hence τc (E ∗ , Ẽ) ≥ γ.

3.25 Proposition (See [Jar81, 11.5.2 p.230]).

Semi-Montel ⇔ quasi-complete and equicontinuous subsets are relatively β(E ∗ , E)-
compact.

Proof. (⇒) semi-Montel ⇒ quasi-complete, β(E ∗ , E) = τpc (E ∗ , E) ⇒ equicontin-

uous sets are relatively β(E ∗ , E)-compact by the Alaŏglu-Bourbaki Theorem 3.4 .
(⇐) By 3.24 and assumption we have τc (E ∗ , Ẽ) = γ(E ∗ , E) ≥ β(E ∗ , E). Let ◦
(resp. •) denote the polarization with respect to the duality (E, E ∗ ) (resp. (Ẽ, E ∗ ))
then for each bounded B ⊆ E there exists a compact K ⊆ Ẽ with K • ⊆ B ◦
and hence (K • )• ⊇ (B ◦ )• ⊇ B. Since the closed absolutely convex hull (K • )• of
(pre)compact sets K is precompact (see the proof of [Kri07b, 6.4.3]) also B is
precompact and by quasi-completeness relatively compact.

3.26 Proposition (See [Jar81, 11.5.4.f p.230]).

Duals of (M)-spaces are (M).

Proof. E ∗ semi-Montel: B ⊆ E ∗ bounded ⇒ B equicontinuous by the uniform

boundedness theorem, since E is barrelled by 3.22 ⇒ B relatively compact for
τpc (E ∗ , E) = β(E ∗ , E) by the Alaŏglu-Bourbaki Theorem 3.4 . Since duals of
reflexive spaces are reflexive they are (infra-)barrelled.

3.27 Proposition (See [Jar81, 11.6.2 p.231]).

A Fréchet space is Montel ((FM) for short) iff it is separable and σ(E ∗ , E)-convergent
sequences are β(E ∗ , E)-convergent.

Proof. (⇒) E (FM) ⇒ E ∗ (M), by 3.26 ⇒ σ(E ∗ , E)-convergent sequences are

convergent, by 3.22 .
{Un : n ∈ N} abs.convex, closed 0-nbhd. basis. We show that EUn is separable,
otherwise ∃ε > 0 ∃A1 ⊆ U1 uncountable with qU1 (x−x0 ) ≥ ε for all x 6= x0 ∈ A1 . U2
is absorbing ⇒ ∃k2 : A1 ∩ k2 U2 uncountable ⇒ . . . ⇒ ∃kn , An : An ⊂ An−1 ∩ kn Un
uncountable. Choose xn ∈ An \ An+1 . Then B := {xn : n ∈ N} is bounded ⇒ B is
relatively compact ⇒ ∃ converging subsequence (xni )i , a contradiction.
3.5
(⇐) U 0-nbhd = ⇒ U o is σ(E ∗ , E)-metrizable ⇒ (U o , σ(E ∗ , E)) → (E ∗ , β(E ∗ , E))
===
is continuous ⇒ U o is β(E ∗ , E)-compact ⇒ E semi-Montel, by 3.25 .

andreas.kriegl@univie.ac.at c July 1, 2016 29

3.28 Montel spaces

3.28 Theorem of Dieudonné-Gommes characterizing Montel for λp (A)

(See [MV92, 27.9 p.310]).
Let A = {a(n) : n ∈ N} be countable. Then

(1) ∃ 1 ≤ p ≤ ∞: λp (A) (M).

⇔ (2) ∀ 1 ≤ p ≤ ∞: λp (A) (M).
⇔ (3) λ∞ (A) = c0 (A).
⇔ (4) λ1 (A) is reflexiv.
⇔ (5) ∀ 1 ≤ p ≤ ∞: not ∃ normed ∞-dim. top.-lin. subspace in λp (A).

⇔ (6) ∀ infinite J ⊆ N ∀n ∃k: inf j∈J aj /aj

(n) (k)
= 0.

2.9.3 2.10
Proof. ( 4 ⇒ 3 ) b ∈ λ∞ (A) = ⇒ W.l.o.g. bj > 0 for all j =====⇒ B := {x :
=====
3.17
kx/bk`1 ≤ 1} is bounded in E := λ1 (A) = ⇒ B is weakly relatively compact in E
====
⇒ ∀k : ιk ◦ ιB : EB  E  Ek weakly compact. Define (for ` := `1 )

R : ` → EB , x 7→ b · x,
S : Ek → `, [x] 7→ x · a(k) , and
D : ` → `, x 7→ b · a(k) · x.

R and S are isometries (by 2.12 and 1.13 ), D = S ◦ ιk ◦ ιB ◦ R : `1 → `1 weakly

compact ⇒ lim b · a(k) = 0, by 3.10.3 , i.e. b ∈ c0 (A).

( 3 ⇒ 2 ) By 3.18 and 2.3 we have to show that bounded sets B in λp (A) are
relatively compact. W.l.o.g. B = Bbp with b ∈ λ∞ (A) by 2.10 . λ∞ (A) = c0 (A) ⇒
D from above (with ` := `p for p < ∞ and ` := c0 for p = ∞) is compact by 3.10
⇒ ∀k : ιk ◦ ιB compact ⇒ B relatively compact (cf. the proof of 3.31 ).

( 2 ⇒ 4 ) By 3.22 Montel spaces are reflexive.

( 1 ⇒ 3 ) As in ( 4 ⇒ 3 ) let b ∈ λ∞ (A) with bj > 0 for all j. Then the bounded

set B := {x : kx/bk`p ≤ 1} is relatively compact in E := λp (A) by ( 1 ). ⇒
∀k : ιk ◦ ιB : EB  E  Ek , x 7→ [x], is compact, where ` := `p for p < ∞
and ` := S(Ek ) ⊆ `∞ for p = ∞. R and S are isometries (by 2.12 and 1.13 ),
D = S ◦ ιk ◦ ιB ◦ R is compact ⇒ lim b · a(k) = 0 by 3.10 , i.e. b ∈ c0 (A).

( 2 ⇒ 1 ) trivial.

( 2 ⇒ 5 ) since normed Montel spaces are finite dimensional by 3.23 and 4.171 .

( 5 ⇒ 6 ) Suppose J ⊆ N, ∃n ∀m ≥ n: inf j∈J aj /aj > 0. ⇒ On E0 := {x ∈

(n) (m)

λp (A) : carr x ⊆ J} the topology induced by k kn coincides with that of λp (A) ⇒

E0 finite dimensional by ( 5 ) ⇒ J finite.
(k)
( 6 ⇒ 3 ) Indirect, suppose ∃b ∈ λ∞ (A) \ c0 (A) ⇒ ∀k ∃Ck > 0 ∀j: |bj |aj ≤ Ck
and ∃n ∃ infinite J ⊆ N ∃ε > 0 ∀j ∈ J: (n)
|bj |aj ≥ ε. ⇒ ∀j ∈ J:
(k)
aj ≤ Ck /|bj | ≤
(n) (n) (k)
Ck aj /ε, i.e. inf{aj /aj : j ∈ J} ≥ ε/Ck , a contradiction.

30 andreas.kriegl@univie.ac.at c July 1, 2016

Montel spaces 3.33

Schwartz spaces

3.29 Definition. Schwartz spaces.

An lcs E is called Schwartz space ((S) for short)(See [MV92, Def. in 24.16
p.265], [Kri07a, 6.4],[Jar81, 10.4 p.201]) iff for each absolutely convex 0-neighborhood
U there exists a 0-neighborhood V ⊆ U such that ιVU : EV → EU is precompact,
S for each ε > 0 exists a finite subset F = {x1 , . . . , xn } ⊆ V with V ⊆ F + ε U =
i.e.
j (xj + ε U ).

3.30 Lemma (See [MV92, 24.17 p.265], [Kri07a, 6.7], [Jar81, 17.1.7 p.370]).
An lcs is Schwartz iff for every continuous linear T : E → F into a normed space
F there exists a 0-neighborhood in E with precompact image in F .

Proof. (⇐) For absolutely convex 0-neighborhoods U consider ιU : E → EU . By

assumption there exists a 0-neighborhood V such that ιU (V ) ⊆ EU is precompact.

(⇒) The set U := T −1 ({x ∈ F : kxk < 1}) is an ab- T /F

E O
solutely convex 0-neighborhood, hence there exists a V
such that ιVU : EV → EU is precompact, so the image ιV T̃
 ιU
!!
T (V ) = T̃ (ιVU (ιV (V ))) is precompact, where T̃ is the con- / / EU
EV
tinuous factorization of T over ιU : E → EU . ιV
U

3.31 Quasi-complete Schwartz implies semi-Montel

(See [MV92, 24.19 p.265], [Jar81, 10.4.3 p.202]).
A Schwartz space is semi-Montel iff it is quasi-complete. (See [Kri07a, 6.6])

Proof. (⇐) Let B ⊆ E be bounded. For every U exists by definition a V with

EV → EU precompact. In particular, ιU (B) is precompact in EU (since V absorbs
B) and hence relatively compact in the completion E fU , see 3.2 . Since Ẽ is com-
Q f Q
plete it is closed in U EU and hence B ⊆ U ιU (B) is by Tychonoffs theorem
relativelv compact in Ẽ, i.e. B ⊆ E is precompact. For the converse use 3.25 .

3.32 Inheritance properties of Schwartz spaces

(See [Kri07a, 6.21], [Jar81, 21.1.7 p.481], resp. [Jar81, 21.2.3 p.483]).
Closed subspaces, products, quotients, and countable coproducts of Schwartz spaces
are Schwartz. This will be shown jointly for nuclear spaces in 3.73 .

3.33 Proposition. (See [Jar81, 10.4.1 p.201], [MV92, 24.22 p.267]).

An lcs E is Schwartz ⇔ ∀U ∃V ⊆ U : U o ⊆ EV∗ o compact, i.e. EU∗ o → EV∗ o is a
compact operator.

Proof. Let U and V be absolutely convex 0-nbhds with V ⊆ U .

U o ⊆ EV∗ o = (EV )∗ is (pre)compact;
o
⇔ (ιVU )∗ = ιU ∗ ∗
V o is τpc ((EU ) , EU )-β((EV ) , EV ))-continuous;
(⇐) U o is τpc ((EU )∗ , EU )-compact by the Alaoğlu-Bourbaki theorem 3.4 ,
hence its image in (EV )∗ is β((EV )∗ , EV )-compact.
(⇒) Since (ιVU )∗ is σ((EU )∗ , EU )-σ((EV )∗ , EV )-continuous and on U o the
topologies τpc ((EU )∗ , EU ) and σ((EU )∗ , EU ) coincide by 3.4 and similarly
on its image (ιVU )∗ (U o ) the topologies β((EV )∗ , EV ) and σ((EV )∗ , EV ) coin-
cide by assumption.
⇔ ∃A ⊆ EU precompact: (ιVU )∗ (A• ) ⊆ ιV (V )o , i.e. (ιU )∗ (A• ) ⊆ V o , where •
denotes the polar with respect to (EU∗ o , EU );

andreas.kriegl@univie.ac.at c July 1, 2016 31

3.36 Schwartz spaces

⇔ ∃A ⊆ EU precompact: ιU (V ) ⊆ (A• )• , see [Jar81, 8.6.2.b p.161];

⇔ ιU (V ) ⊆ EU precompact, by the bipolar theorem.

3.34 Proposition (See [Jar81, 11.6.3 p.232]).

A Fréchet space is Schwartz space ((FS) for short) iff it is separable and σ(E ∗ , E)-
convergent sequences are equicontinuously convergent, i.e. uniformly convergent on
some 0-neighborhood V , or, with other words, convergent in the normed space EV∗ o .

Proof. (⇒) (FS) ⇒ (M) ⇒ separable, by 3.31 , 2.3 , and 3.27 .

Let (un ) be σ(E ∗ , E)-convergent ⇒ U := {un : n ∈ N}o ⊆ E is a barrel, hence a
0-nbhd ⇒ ∃V 0-nbhd. with ({un : n ∈ N}o )o compact in EV∗ o = (EV )∗ , by 3.33
⇒ un converges equicontinuously.
(⇐) Let U 0-nbhd. and U ⊇ Uk ⊇ Uk+1 be a 0-nbhd. basis. By 3.33 we have
to show ∃k : U o ⊆ EU∗ o is compact. Since U o is σ(E ∗ , E)-metrizable, by 3.5 , it
k
suffices to show that there exists some k such that σ(E ∗ , E)-converging sequences
in U o converge in Fk := EU∗ o . Otherwise, ∀k ∃(ukn )n convergent in U o (towards
k
0) but not convergent in Fk for n → ∞. Let B0 ⊇ B1 ⊇ . . . be a countable 0-
nbhd. basis for the metrizable topology σ(E ∗ , E) on U o . Let mn := min{m : uki ∈
Bn ∀k ≤ n ∀i > m}, then u1mn +1 , . . . , u1mn+1 ; . . . ; unmn +1 , . . . , unmn+1 ∈ Bn . These
blocks together give a weak 0-sequence (un ) in U o , not convergent in Fk , (i.e. not
equicontinuously convergent), since (ukn )n>mk is a subsequence, a contradiction.

3.35 Theorem. Characterizing Schwartz for λp (A)

(See [MV92, 27.10 p.312]).
Let A = {a(k) : k ∈ N} be countable.
(1) ∃ 1 ≤ p ≤ ∞: λp (A) (S)
⇔ (2) ∀ 1 ≤ p ≤ ∞: λp (A) (S)
(k) (m)
⇔ (3) ∀k ∃m ≥ k: limj→∞ aj /aj = 0.

Proof. Let E = λp (A) for 1 ≤ p ≤ ∞, Ek ∼

= `p (ak ) for 1 ≤ p < ∞ (cf. 1.13 ),
w.l.o.g. carr ak = N. For m ≥ k define
D : x 7→ x · ak /am , `p → `p
Am : x 7→ x · am , Em → `p
Am is isometry and D = Ak ◦ ιkm ◦ A−1 p
m . If p = ∞ replace ` by ` := Am (Em ).

( 1 ⇒ 3 ) ⇒ ∀k ∃m ≥ k: ιkm : Em → Ek compact ⇒ D = Ak ◦ ιkm ◦ A−1

m compact
3.10
= ⇒ lim a(k) /a(m) = 0.
====
3.10
( 3 ⇒ 2 ) Let 1 ≤ p ≤ ∞. ∀k ∃m ≥ k satisfying ( 3 ) = ⇒ D = Ak ◦ ιkm ◦ A−1
==== m
compact ⇒ ιkm compact ⇒ ( 2 ).
( 2 ⇒ 1 ) trivial.

3.36 Example of (FM), but not Schwartz (See [MV92, 27.21 p.319]).
(
(ki)k for j < k
A := {a : k ∈ N} with ai,j :=
(k) (k)
⇒ λp (A) is (F), (M), not (S)
kj for j ≥ k

(k) (m) 3.35

Proof. m > k, j > m ⇒ ai,j /ai,j = (k/m)j = ⇒ λp (A) not Schwartz.
====

32 andreas.kriegl@univie.ac.at c July 1, 2016

Schwartz spaces 3.37

Let I ⊆ N2 , n fixed. ∀k ≥ n: inf (i,j)∈I ai,j /ai,j =: εk > 0.

(n) (k)

Claim: I is finite:
k := n + 1, j ≥ n + 1 ⇒
 j
(n) (n+1) n
εn+1 ≤ ai,j /ai,j =
n+1
⇒ ∃j0 : I ⊆ N × {1, . . . , j0 }. Let 1 ≤ j ≤ j0 , k > max{j, n}, (i, j) ∈ I ⇒
(ni)n nn n−k
 (n) (k)
ai,j /ai,j = (ki)k = kk i
 for j < n(< k)
εk ≤
 (n) (k) nj

ai,j /ai,j = (ki) k for (k >)j ≥ n
⇒ I ∩ (N × {j}) is finite.
3.28.6
⇒ λp (A) Montel (for all 1 ≤ p ≤ ∞).
======

Tensor products

In this section we introduce the projective tensor product as universal solution for
linearizing bilinear continuous maps and the injective tensor product as subspace
of te space of all bounded linear (or bilinear) operators. of locally convex spaces
in order to define nuclearity. Nuclear spaces are then deinied as those locally
convex spaces, where these to tensor product functors coincide. And we use these
tensor products to obtain descriptions for various types of vector valued summable
sequences.

3.37 Definition. Projective tensor product (See [Kri07a, 3.3 p.53]).

The algebraic tensor product E ⊗ F of two linear spaces E and F is the
universal solution for turning bilinear mappings into linear ones, i.e. there exists a
bilinear mapping ⊗ : E × F → E ⊗ F such that

E×F
⊗
/ E⊗F

∃!f˜ linear
∀f bilinear % 
∀G
The linear space E ⊗ F can be obtained as subspace of L(E, F ; K)∗ (the dual of
the bilinear forms) generated by the image of ⊗ : E × F → E ⊗ F ⊆ L(E, F ; K)∗
given by (x, y) 7→ ev(x,y) (See [Kri07a, 3.1 p.50]).
For locally convex spaces the solution of the corresponding universal problem for
(bi)linear continuous mappings is called projective tensor product E ⊗π F ,
it is the linear spaces E ⊗ F supplied with the finest locally convex topology for
which ⊗ : E × F → E ⊗ F is continuous. This topology exists since the union of
locally convex topologies is locally convex and E × F → E ⊗ F is continuous for the
weak topology on E ⊗ F generated by those linear functionals which correspond
to continuous bi-linear functionals on E × F . It has the universal property, since
the inverse image of a locally convex topology under a linear mapping T̃ is again a
locally convex topology, such that ⊗ is continuous, provided the associated bilinear
mapping T is continuous.
The space E ⊗πPF is Hausdorff, since the set E ∗ × F ∗ separates points in E ⊗ F :
Let 0 6= z = k xk ⊗ yk be given. By replacing linear dependent xk by the
corresponding linear combinations and using bilinearity of ⊗, we may assume that

andreas.kriegl@univie.ac.at c July 1, 2016 33

3.39 Tensor products

the xk are linearly independent. Now choose x∗ ∈ E ∗ and y ∗ ∈ F ∗ be such that

x∗ (xk ) = δ1,k and y ∗ (y1 ) = 1. Then (x∗ ⊗ y ∗ )(z) = 1 6= 0.
We denote the space of continuous linear mappings from E to F by L(E, F ),
and the space of continuous multi-linear mappings by L(E1 , . . . , En ; F ).
ˆ πF .
The completion of E ⊗π F will be denoted E ⊗
Since a bilinear mapping is continuous iff it is so at 0, a 0-neighborhood basis in
E⊗π F is given by all those absolutely convex sets, for which the inverse image under
⊗ is a 0-neighborhood in E ×F . A basis is thus given by the absolutely convex hulls
denoted U ⊗ V of the images of U × V under ⊗, where U resp. V runs through a
0-neighborhood basis of E resp. F . We only have Pto show that these sets U ⊗ V are
absorbing (see [Jar81, 6.5.3 p.108]). So let z = k≤K xk ⊗yk ∈ E ⊗F be arbitrary.
ThenPthere are ak > 0 and P bk > 0 such that xk ∈ ak U and yk ∈ bk V and hence
z = k ak bk xakk ⊗ ybkk ∈ ( k ak bk ) · hU ⊗ V iabs.conv. . Consequently, the Minkowski-
functionals pU ⊗V form a base of the seminorms of E ⊗π F and we will denote them
by πU,V . In
P terms of the Minkowski-functionals pP U and pV of U and V we obtain
that z ∈ ( k pU (xk ) pV (yk )) U ⊗ V forP any z = k xk ⊗ yk since Pxk ∈ pU (xk ) · U
for closed U , and thus pU ⊗V (z) ≤ inf p
k U (x k ) p V (yk ) : z = k xk ⊗ yk . We
now show the converse:

3.38 Proposition. Seminorms of the projective tensor product

(See [Kri07a, 3.4 p.53], [Jar81, 15.1.1 p.324]).
nX X o
pU ⊗V (z) = inf pU (xk ) · pV (yk ) : z = xk ⊗ yk .
k k

P
Proof. Let z ∈ P λ · (U ⊗ V ) with λ > 0. Then P z = λ λk (uk ⊗ vk ) with uk ∈
P, vk ∈ V and
U k |λ
Pk | = 1. Hence z = xk ⊗ vk , where xk = λλk uk , and
k p U (x k )·pV (v k ) ≤ λ|λ k | = λ. Taking the infimum of all λ shows that pU ⊗V (z)
is greater or equal to the infimum on the right side.

3.39 Theorem. Compact subsets of the projective tensor product

(See [Kri07a, 3.21 p.61] and [Jar81, 15.6.3 p.336]).

Compact subsets of the completed projective tensor product E ⊗ ˆ π F for metrizable

spaces E and F are contained in the closed absolutely convex hull of a tensor product
of precompact sets in E and F .

Proof. Every compact set K in the Fréchet space E ⊗ ˆ π F is contained in the closed
ˆ π F by 3.6 . For this 0-sequence
absolutely convex hull of a 0-sequence (zk )k in E ⊗
we can choose kn strictly increasing, such that zk ∈ Un ⊗ Vn for all k ≥ kn , where
(Un )n and (Vn )n are countable 0-neighborhood bases of the topology P of E and F .
For kn ≤ k < kn+1 we can choose finite (disjoint) sets Nk ⊆ N and j∈Nk |λj | = 1,
P F
xj ∈ Un and yj ∈ Vn such that zk = j∈Nk λj xj ⊗ yj . Let A := {xj : j ∈ k Nk }
F
and B := {yj : j ∈ k Nk }. These are formed by two sequences converging to 0,
and hence are precompact. Furthermore, each z ∈ K can be written as
∞
X X X
z= µk zk = µk λj xj ⊗ yj
k=0 k j∈Nk
P P P P P
with k |µk | ≤ 1 and hence k j∈Nk |µk λj | = k |µk | j∈Nk |λj | ≤ 1. From
this it easily follows that the series on the right hand side converges (even Mackey)
and hence z is contained in the closed absolutely convex hull of A ⊗ B.

34 andreas.kriegl@univie.ac.at c July 1, 2016

Tensor products 3.41

3.40 Corollary. Elements of the completed tensor product as limits

(See [Kri07a, 3.22 p.61], [Jar81, 15.6.4 p.337]).
For z ∈ E⊗ˆ π F has a representation of the form z =
P metrizable E and F every 1
n λ n x n ⊗ y n , where λ ∈ ` and x and y are bounded (or even 0-)sequences.

Since for every λ ∈ `1 there exists a ρ ∈ c0 and µ ∈ `1 with λn = ρ2n µn it is enough

to find bounded sequences xn and yn .
P
Proof. In the previous proof we have just shown that z = j µkj λj xj ⊗ yj .

3.41 Definition. Summable vector valued sequences.

For lcs F we consider the following spaces of (somehow summable) series in F :
• `1 {F } := `1 (N, F ) := f ∈ F N : ∀p SN of F : p̃(f ) := j=0 p(fj ) < ∞ ,
 P∞

the space of absolutely summable sequences in F (called absolutely

Cauchy sequences in [Jar81, 15.7.5 p.341]).
Recall the Reordering Theorem of Riemann [Kri05, 2.5.18].
• `1 hF i, the space of unconditionally Cauchy summable sequences
(xj )j∈N in F (see [Jar81, 14.6.1 p.305]), i.e. for which the net F 7→ j∈F xj ,
P
where F runs through the finite subsets of N ordered by inclusion, is Cauchy:
P Let σ bePa permutation of N. For any U we find a finite F0 ⊆ N such
(⇐)
−1
that
k∈F2 kx − x
k∈F1 k ∈ U for all finite F 1 , F2 ⊇ F 0 . Let n 0 := max σ (F 0 ),
−1
hence σ (F0 ) ⊆ {n : n ≤ n0 }. Then, for all n2 ≥ n1 > n0 , we have
n2
X X X X X
xσ(n) = xσ(n) − xσ(n) = xk − xk ∈ U, where
n=n1 n≤n2 n<n1 k∈F2 k∈F1
F2 := σ({n : n ≤ n2 }) ⊇ F1 := σ({n : n < n1 }) ⊇ σ({n : n ≤ n0 }) ⊇ F0 .

0
finite : F 0 ∩P
P
(⇒) Otherwise, ∃U ∀F P finite ∃F P F = ∅ and n∈F
P 0 xn ∈/ 2U
(∃F1 , F2 ⊇ F : 4U 63 n∈F2 xn − n∈F1 xn = n∈F2 \F1 xn − n∈F1 \F2 xn ,
now take F 0 := F2 \P F1 or F 0 := F1 \ F2 ). Since n xn is Cauchy, there is
P
n2
some n0 such that n=n xn ∈ U for all n2 ≥ n1 ≥ n0 . Let F0 := {n ∈
N : n ≤ n0 } and F0 a corresponding set. We construct nk , Fk , and Fk0 6= ∅
1
0

recursively as nk+1 := max Fk0 , Fk+1 := {n : n ≤ nk+1 } ⊇ Fk0 ∪ Fk . Let

Fk00 := Fk+1 \ (Fk t Fk0 ). Then
X X X X X X
xn = xn − xn − xn = xn − xn ,
n∈Fk00 n∈Fk+1 n∈Fk n∈Fk0 n∈Fk+1 \Fk n∈Fk0

where
P Fk+1 \ Fk = {n : nkP< n ≤ nk+1 } with nP k+1 ≥ nk ≥ n0 , so
n∈Fk+1 \Fk x n ∈ U , wheras n∈Fk 0 x n ∈/ 2U , hence n∈Fk00 xn ∈
/ U . The
elements inP the sequence F0 , F0
0
, F 00
0 , F1
0
, F 00
1 , . . . define a permutation σ of N
for which n xσ(n) is not Cauchy.
• `1 [F ] := L(c0 , F ), the space of scalarly absolutely summable se-
quences in F (See [Kri07a, 4.9] and [Jar81, 19.4.3 p.427]): Since the
standard unit vectors ek generate a dense subspace in c0 every f ∈ L(c0 , F )
is uniquely determined by its values P fk := f∗ (ek ). Moreover, f is contin-
uous=bounded iff {(y ∗ ◦ f )(x) = j∈N xj y (fj ) : x ∈ c0 , kxk∞ ≤ 1} is
∗ ∗ ∗
bounded for each y ∈ F , i.e. {(xj y (fj ))j : x ∈ c0 , kxk∞ ≤ 1} is bounded
in `1 , i.e. (y ∗ (fj ))j ∈ λ1 (c0 ) = `1 by 1.15.7 , i.e. (fj )j is scalarly absolutely
summable.
This can be extended 1 < q < ∞:

andreas.kriegl@univie.ac.at c July 1, 2016 35

3.43 Tensor products

• `q {F } := `q (N, F ) := f ∈ F N : ∀p : p̃(f ) := q 1/q

 P∞

j=0 p(fj ) < ∞ ,
the space of absolutely q-summable sequences in F (See [Jar81, 19.4
p.425]).
• `q [F ] = L(`p , F ), the space of scalarly absolutely q-summable se-
quences in F , where p1 + 1q = 1 (See [Jar81, 19.4.1 p.426] and [Jar81,
19.4.3 p.427]): Since the standard unit vectors ek generate a dense subspace
in `p every f ∈ L(`p , F ) is uniquely determined by itsP values fk := f (ek ).
Moreover, f is continuous=bounded iff {(y ∗ ◦ f )(x) = j∈N xj y ∗ (fj ) : x ∈
`p , kxkp ≤ 1} is bounded for each y ∗ ∈ F ∗ , i.e. {(xj y ∗ (fj ))j : x ∈ `p , kxkp ≤
1} is bounded in `1 , i.e. (y ∗ (fj ))j ∈ λ1 (`p ) = `q by 1.15.7 , i.e. (fj )j is
scalarly absolutely q-summable.

3.42 Lemma. Description of `1 {F } as tensor product

(See [Kri07a, 4.12], [Jar81, 15.7.6 p.341]).
For lcs F we have a dense topological embedding `1 ⊗π F ,→→ `1 (N, F ).
Thus ` ⊗π F ∼
1ˆ
= ` (N, F ) for complete F , where ⊗
1 ˆ π denotes the completion of the
projective tensor product.

Proof. We first show that the natural mapping `1c ⊗π F → `1c (N, F ), x ⊗ y 7→
(xj y)j∈N , is an isomorphism, where `1c is the dense subspace in `1 of finite sequences
and `1c (N, F ) the analogous
P (k) subspace in `1 (N, F ). Since Rk ⊗π F ∼ = F k we have a
(k) 1
bijection. Let z = k x ⊗ y ∈ `c ⊗ F and p be a seminorm of F . For the
corresponding norm p̃ of `1c (N, F ) we have
!
X X X (k) X X (k)
(k)
p̃(z) := p(zj ) = p xj y ≤ |xj | p(y (k) ) ≤
j j k j k
(k)
XX X
≤ |xj | p(y (k) ) = kx(k) k`1 · p(y (k) ),
k j k

Taking the infimum of the right side over all representations of z shows that p̃ ≤ pπ ,
where pπ is projective tensor norm formed from k k`1 and p, see 3.38 .
Conversely each z = (zj )j ∈ `1c (N, F ) can be written as image of the finite sum
1
P
j ej ⊗ zj , where ej denotes the standard unit vector in ` . Thus we have for the
π
tensor norm p that
X X
pπ (z) ≤ kej k`1 · p(zj ) = p(zj ) = p̃(z)
j j

which shows the converse relation.

Now, since `1c (N, F ) is dense in `1 (N, F ) and the latter space is complete for com-
plete F (as can be shown analogously to the case `1 (N, R)), we have the desired
isomorphism:
`1 (N, F ) = `1c\
(N, F ) ∼ ˆ πF ∼
= `1c ⊗ = `1 ⊗
ˆ π F.
Here we used that the dense emnbedding `1c ,→ `1 induces a dense embedding
`1c ⊗π F ,→ `1 ⊗π F , see [Kri07a, 3.19,3.20] or [Jar81, 15.2.3,15.2.4 p.327].

3.43 Lemma. The seminorms of `1 [F ] (See [Kri07a, 4.32], [Jar81, 19.4.3a

p.427]).
The structure on `q [F ] induced from L((`q )∗ , F ) for 1 < q < ∞ (resp. from L(c0 , F )
for q = 1) is given by the seminorms
∞
n X 1/q o
p̃(f ) := sup |y ∗ (fn )|q : |y ∗ | ≤ p ,
n=1

36 andreas.kriegl@univie.ac.at c July 1, 2016

Tensor products 3.44

where p runs through all continuous seminorms of F .

Proof. Let p be a continuous seminorm on F and V := {y ∈ F : p(y) ≤ 1}. As

in 3.45 we use that p(y) = sup{|y ∗ (y)| : y ∗ ∈ V o }. Thus we can calculate the
seminorm p∞ on L((`q )∗ , F ) associated to p as follows, where B denotes the closed
unit-ball in ` := (`q )∗ (resp. c0 for q = 1) and ι : `q [F ] → L(`, F ), ι(f )(λ) :=
P
k fk λk , the canonical bijection:

p̃(f ) := p∞ (ι(f )) := sup{p(ι(f )(λ)) : λ ∈ B}

n X ∞  o
= sup y ∗ fk λk : λ ∈ B, y ∗ ∈ V o
k=1
n o
≤ sup kλk(`q )∗ k(y ∗ (fk ))k k`q : λ ∈ B, y ∗ ∈ V o
| {z }
≤1
∞
nX 1/q o
≤ sup |y ∗ (fk )|q : |y ∗ | ≤ p
k=1
q ∗
Conversely, let f ∈ ` [F ] and |y | ≤ p. Then for ε > 0 there exists an n such that
∗ q 1/q
< ε. Let λk y ∗ (fk ) := |y ∗ (fk )| for k ≤ n and λk = 0 otherwise.
P

k>n |y (fk )|
Then λ ∈ B and
∞
X X X ∞
X
|y ∗ (fk )|q = (λk y ∗ (fk ))q + |y ∗ (fk )|q ≤ (λk y ∗ (fk ))q +εq ≤ p∞ (ι(f ))q +εq .
k=1 k≤n k>n k=0

Hence we have also the converse relation.

3.44 Definition. Injective tensor product

(See [Kri07a, 4.21 p.93], [Jar81, 16.1 p.344]).
We consider the bilinear mapping
E × F → L(E ∗ , F ), given by (x, y) 7→ x∗ 7→ x∗ (x)y .

It is well-defined, since evx : E ∗ → R is bounded. In fact evx : E ∗ → R is even

continuous for the weak topology σ(E ∗ , E) and hence also for the topology β(E ∗ , E)
of uniform convergence on bounded sets. This induces a linear map
E ⊗ F → L(E ∗ , F ), given by x ⊗ y 7→ (x∗ 7→ x∗ (x)y).
P
We claim that this mapping is injective. In fact take i xi ⊗ yi ∈ E ⊗ F with xi
linearly independent. By Hahn-Banach we can find P continuous linear ∗functionals
x∗i with x∗i (xj ) = δi,j . Assume that the image of i xi ⊗ yi isP 0 in L(E , F ). Since
it has value yi on x∗i , we have that yi = 0 for all i and hence i xi ⊗ yi = 0.
We define the injective tensor product (also called ε-tensor product in
[Tre67]) E ⊗ε F to be the algebraic tensor product with the locally convex topology
induced by the injective inclusion into L(E ∗ , F ), where L(E ∗ , F ) is supplied with
the topology of uniform convergence on equicontinuous subsets of E ∗ . Since this
topology on L(E ∗ , F ) is obviously Hausdorff, the same is true for E ⊗ε F .
Note that, since F topologically embeds into the space (F ∗ )0 of bounded (with
respect to the equicontinous subsets of E ∗ ) linear functionals on E ∗ by [Kri14,
5.4.11], the structure of E ⊗ε F is also initial with respect to E ⊗ F
→ L(E ∗ , F ) →
= L(E ∗ , F ∗ ; R), x ⊗ y 7→ (x∗ , y ∗ ) 7→ x∗ (x) · y ∗ (y) , which gives a
L(E ∗ , (F ∗ )0 ) ∼
more symmetric form and consequently E ⊗ε F ∼ = F ⊗ε E. Since the seminorms of

andreas.kriegl@univie.ac.at c July 1, 2016 37

3.49 Tensor products

L(E ∗ , F ∗ ; R) are given by suprema on U o ×V o , where U and V are 0-neighborhoods,

we have for the corresponding seminorm εU,V on E ⊗ε F :
X  nX o
εU,V xk ⊗ yk := sup x∗ (xk ) y ∗ (yk ) : x∗ ∈ U o , y ∗ ∈ V o
k k

3.45 Corollary. Seminorms of the injective tensor product

(See [Kri07a, 4.22 p.94], [Jar81, 16.1 p.344]).

A defining family of seminorms on E ⊗ε F is given by

X nX o
εU,V : xi ⊗ yi 7→ sup x∗ (xi ) y ∗ (yi ) : x∗ ∈ U o , y ∗ ∈ V o ,
i i
where U and V run through the 0-neighborhoods of E and F . The injective tensor
product E ⊗ε F is metrizable (resp. normable) if E and F are.
Let us show next, that the canonical bilinear mapping E × F → L(E ∗ , F ) is con-
tinuous, which implies that the identity E ⊗π F → E ⊗ε F is continuous:
In fact, take an equicontinuous set E ⊆ E ∗ , i.e. E is contained in the polar U o of
a 0-neighborhood U , and take furthermore an absolutely convex 0-neighborhood
V ⊆ F . Then U × V is mapped into the typical 0-nbhd. {T : T (E) ⊆ V }, since
(x ⊗ y)(x∗ ) = x∗ (x) y ∈ {λ : |λ| ≤ 1} · V ⊆ V for x∗ ∈ E ⊆ U o .

3.46 Corollary (See [Kri07a, 4.23 p.94], [Jar81, 16.1.3 p.345]).

E ⊗π F → E ⊗ε F is continuous.

Proof. In the diagram

E ⊗O π F / E ⊗ε F
_
⊗

E×F / L(E ∗ , F )
continuity of the bilinear map at the bottom implies continuity of the top arrow.

3.47 Definition (See [Kri07a, 4.24 p.94], [Jar81, 16.1.4 p.345]).

An lcs E is called nuclear ((N) for short) iff E ⊗π F = E ⊗ε F for all lcs F .

3.48 Corollary (See [Kri07a, 4.26]).

The space E 0 ⊗ε F embeds into L(E, F ).

Proof. In fact, since obviously E 0 ⊗ε F = ∼ F ⊗ε E 0 , it embeds into L(F ∗ , E 0 ) ∼

=
L(E, (F ) ) via x ⊗ y 7→ (x 7→ (y 7→ x (x) y ∗ (y))). This embedding factors over
∗ 0 ∗ ∗ ∗

the embedding δ∗ : L(E, F ) ,→ L(E, (F ∗ )0 ), by x∗ ⊗ y 7→ (x 7→ x∗ (x)y). Hence this

map E 0 ⊗ε F → L(E, F ) is an embedding.
 / L(E, (F ∗ )0 ) / x 7→ (y ∗ 7→ x∗ (x) y ∗ (y))
E 0 ⊗ε F s x∗ ⊗ y



O  O
δ∗ δ∗
& ? ( _

L(E, F ) (x 7→ x∗ (x)y)

3.49 Lemma. Completeness of `1 hF i

(See [Kri07a, 4.33], [Jar81, 16.5.1 p.358]).
The subspace `1 hF i of `1 [F ] is closed. For complete F both spaces are complete.
Hence we will always consider the initial structure on `1 hF i induced from `1 [F ].

38 andreas.kriegl@univie.ac.at c July 1, 2016

Tensor products 3.50

Proof. In order to show that `1 hF i is closed in `1 [F ],P

take x = (xk )k ∈ `1 [F ] in the
1
closure of ` hF i. We have to show that the net K 7→ k∈K xk is Cauchy, where K
runs through the finite subsets of N. So let p be a seminorm of F and εP > 0. By
the assumption we can find a y ∈ `1 hF i with p̃(x − y) ≤ ε. Thus the net k∈K
yk
is Cauchy in F , i.e. there is a finite K0 ⊆ N such that p
P P
y
k∈K2 k − k∈K1 k ≤ ε
y
for all K1 , K2 ⊇ K0 . Hence for K0 ⊆ K1 ⊂ K2 we have:
X X  X  X  X 
p xk − xk = p xk ≤ p (xk − yk ) + p yk
k∈K2 k∈K1 k∈K2 \K1 k∈K2 \K1 k∈K2 \K1
n X  o X 
≤ sup y ∗ (xk − yk ) : |y ∗ | ≤ p + p yk
k∈K2 \K1 k∈K2 \K1
∞
nX o X 
≤ sup |y ∗ (xk − yk )| : |y ∗ | ≤ p + p yk
k=0 k∈K2 \K1

3.43 X X 
= p̃(x − y) + p
====
= yk − yk ≤ ε + ε,
k∈K2 k∈K1
P
which shows that K 7→ k∈K xk is a Cauchy-net.

Since `1 [F ] ∼
= L(c0 , F ), it is complete for complete F .

3.50 Theorem. Description of `1 hF i as tensor product

(See [Kri07a, 4.34], [Jar81, 16.5.2 p.359]).
For lcs F we have a dense topological embedding `1 ⊗ε F ,→
→ `1 hF i.
Thus ` ⊗ε F ∼
1b 1 ˆ ε denotes the completion of the
= ` hF i for complete F , where ⊗
injective tensor product.

Proof. By 3.48 we have that `1 ⊗ε F ∼ = c00 ⊗ε F embeds into L(c0 , F ), the space
of scalarly absolutely summable sequences. Obviously λ ⊗ y ∈ `1 ⊗ F is contained
in `1 {F } ⊆ `1 hF i. We show that `1c ⊗ F = K(N) ⊗ F ∼ = F (N) is dense in `1 hF i
with respect to the structure inherited from ` [F ]. So let x ∈ `1 hF i and consider
1

xn := x|[0,...,n−1] ∈ F n ⊆ F (N) ⊆ `1 [F ]. We n 1
P claim that x → x in ` [F ]: Let p be a
continuous seminorm on F . Since K 7→ k∈K xk is Cauchy, we have for K = R:
m
3.43 nX o nX o
p̃(x − xn ) =
====
= sup |y ∗ (xk )| : |y ∗ | ≤ p = sup |y ∗ (xk )| : |y ∗ | ≤ p, m ≥ n
k≥n k=n
n X X o
= sup y ∗ (xk ) + y ∗ (xk ) : |y ∗ | ≤ p, m ≥ n
m≥k≥n m≥k≥n
y ∗ (xk )>0 y ∗ (xk )<0
n X  o n X  o
≤ sup y ∗ xk : |y ∗ | ≤ p, m ≥ n + sup y ∗ xk : . . .
m≥k≥n m≥k≥n
y ∗ (xk )>0 y ∗ (xk )<0
n X  o
≤ 2 sup p xk : K 0 finite, K 0 ∩ [0, n − 1] = ∅ ≤ 2ε
k∈K 0

for n sufficiently
P large. In the complex case we have to make a more involved
estimation for k>n |y ∗ (xk )|. Let P := {z ∈ C : <z > 0 and − <z < =z ≤ <z}.
For every z 6= 0 there is a unique j ∈ {0, 1, 2, 3} with ij z ∈ P . Then |z| ≤ 2<(ij z) ≤
2|z|. Thus we can split the sum into 4 parts corresponding to j ∈ {0, 1, 2, 3}, where

andreas.kriegl@univie.ac.at c July 1, 2016 39

3.53 Tensor products

ij y ∗ (xk ) ∈ P . For each subsum we have

X X   X 
|y ∗ (xk )| = 2<(ij y ∗ (xk )) = 2 < ij y ∗ xk
m≥k≥n m≥k≥n m≥k≥n
ij y ∗ (xk )∈P ij y ∗ (xk )∈P ij y ∗ (xk )∈P
 X   X 
≤ 2 y∗ xk ≤ 2p xk ≤ 2 ε
m≥k≥n m≥k≥n
ij y ∗ (xk )∈P ij y ∗ (xk )∈P

Thus we have p̃(x − xn ) ≤ 8 ε.

Since `1 hF i is complete for complete F by 3.49 , the result follows.

Operator ideals

In order to give several equivalent descriptions of nuclear spaces in terms of the

connecting morphism in their projective representation, we introduce the ideals of
approximable, of nuclear, and of summing operators between Banach spaces and
prove the most relevant relations between them.
In this section all lcs are assumed to be Banach spaces!

3.51 Definition. Several operator ideals.

For 1 ≤ p < ∞ define the following classes of operators between Banach spaces:
• Ap , the class of p-approximable operators (See [Kri07a, Def. before
5.26 p.128]), i.e. those for which the approximation numbers (an (T )) ∈ `p ,
see 3.11 . WARNING: This class is denoted Sp (for Schatten-class) in
[Jar81, 19.8 p.440] and [MV92, 16.6 p.143]!
• Np , the class of p-nuclear
P∞ operators, i.e. those which have a representa-
tion of the form T = n=0 x∗n ⊗ yn with (x∗n ) ∈ `p {E ∗ } and (yn ) ∈ `q {F },
where p1 + 1q = 1, see [Kri07a, 5.9].
• Sp , the class of p-summing operators, i.e. those with T∗ (`p [E]) ⊆ `p {F },
see [Kri07a, 5.18]. These classes are denotes Pp in [Jar81, 19.5 p.428].
In the case p = 1 we suppress the “1-” from these definitions.
P In particular, T is a
nuclear operator, iff there exists aj ∈ E ∗ and bj ∈ F with j kaj k kbj k < ∞ and
X
T (x) = aj (x) bj for all x.
j∈N

All these classes are operator ideals, since for A, B ∈ L they are closed under
T 7→ A ◦ T ◦ B. For approximable this follows from an+m (R ◦ S) ≤ an (R) · am (S),
see 3.53 below, for the others from S∗ (`p {E}) ⊆ `p ({F }) and S∗ (`p [E]) ⊆ `p [F ]
(since `q (N, ) and L(`, ) are obviously functorial).

3.52 Lemma (See [Jar81, 17.3.3 p.377]).

The space N1 (E, F ) of nuclear operators is the image of E ∗ ⊗
ˆ π F in L(E, F ).

Proof. By 3.40 the elements of E ∗ ⊗ ˆ π F are those of the from λn x∗n ⊗ yn with
P
n
x∗ , y bounded sequences and λ ∈ `1 .

3.53 Proposition (See [Kri07a, 5.29], [Jar81, 19.10.1 p.445]).

Let 0 < p, q, r < ∞ with 1r = p1 + 1q .
Then Aq ◦ Ap ⊆ Ar . In particular, we will use A2 ◦ A2 ⊆ A1 and (A1 )n ⊆ A1/n .

40 andreas.kriegl@univie.ac.at c July 1, 2016

Operator ideals 3.55

Proof. We have an+m (S ◦ T ) ≤ an (S) am (T ):

In fact, let R := S0 ◦ T + (S − S0 ) ◦ T0 for n- resp. m-dimensional S0 resp. T0 , then
an+m (S ◦ T ) ≤ kS ◦ T − Rk = k(S − S0 ) ◦ (T − T0 )k ≤ kS − S0 k · kT − T0 k.
r r
Using the Hölder inequality for p + q = 1 we obtain:
X 1/r X 1/r X 1/r
an (S ◦ T )r ≤ 21/r a2n (S ◦ T )r ≤ 21/r an (S)r · an (T )r
n n n
X 1/p X 1/q
1/r p
≤2 an (S) · an (T )q
n n

3.54 Proposition (See [Kri07a, 5.30], [Jar81, 20.2.3 p.454]).

An linear operator T : E → F between Hilbert spaces is p-approximable provided
(hT en , fn i)n ∈ `p for all orthonormal sequences en and fn .

It can be shown that the converse is valid as well, see [Jar81, 20.2.3 p.454].

Proof. By 3.9 we conclude

P that T is compact and hence admits by 3.8 a
representation T x = n n n , xifn with λn → 0 and orthonormal sequences
λ he
en and fn . Since λn = hT en , fn i we have that (λn )n ∈ `p . By applying a
P and putting signs to fn we may assume that 0 < λn+1 ≤ λn Let
permutation
Tn (x) := k<n λk hek , xifk . Then
n X o
an (T ) ≤ kT − Tn k = sup λk hek , xifk : kxk ≤ 1
k≥n
n X 1/2 o
= sup λ2k |hek , xi|2 : kxk ≤ 1 ≤ λn ,
k≥n

hence T ∈ Ap .

3.55 Auerbach’s Lemma (See [Kri07a, 5.26], [Jar81, 14.1.7 p.291]).

Let E be a finite dimensional Banach space. Then there are unit vectors xi ∈ E
and x∗i ∈ E ∗ with x∗i (xj ) = δi,j for 1 ≤ i, j ≤ dim E.

Proof. Let e1 , . . . , en be an algebraic basis of E. For the weakly compact unit ball
K of E ∗ we consider the continuous map f : K n → K, (x∗1 , . . . , x∗n ) 7→ | det(x∗j (ei ))|.
Let (x∗1 , . . . , x∗n ) be a point where it attains its maximum. Since the ei are linearly
independent this maximum is positive. Hence there is a unique solution with xj ∈ E
of the equations
X
x∗j (ei )xj = ei for 1 ≤ i ≤ n.
j

Applying any x∗k to this equation, yields the equations

X
x∗j (ei ) x∗k (xj ) = x∗k (ei ) for 1 ≤ i ≤ n.
j

whose unique solution is x∗j (xi ) = δi,j .

f (x∗1 , . . . , x∗n ) · det(yj∗ (xi )) = det(x∗j (ei )) · det(yj∗ (xi ))

X 
= det x∗k (ei ) yj∗ (xk ) = det(yj∗ (ei ))
k
= f (y1∗ , . . . , yn∗ ) ≤ f (x∗1 , . . . , x∗n ) for all yi∗ ∈ K.

andreas.kriegl@univie.ac.at c July 1, 2016 41

3.59 Operator ideals

Thus | det(yj∗ (xi ))| ≤ 1. Choosing yj∗ = x∗j for all j 6= k shows that |yk∗ (xk )| ≤ 1
and hence kxk k ≤ 1. From 1 = x∗j (xj ) ≤ kx∗j k kxj k we conclude that kxj k = 1 =
kx∗j k.

3.56 Lemma. (See [Kri07a, 5.27], [Jar81, 19.8.4 p.441]).

Let T ∈ L(E, F ) be such that dim T (E) = k < ∞. Then T can be written as
Pk
T = j=1 λj x∗j ⊗ yj with kx∗j k ≤ 1 and kyj k ≤ 1 and 0 < λj ≤ kT k.

Proof. We may assume that T is onto. By 3.55 we have a biorthogonal sequence

yj and yj∗ for F . Let λj := kT ∗ yj∗ k. Then 0 < λj ≤ kT ∗ k = kT k and x∗j :=
∗ ∗ ∗
1
P ∗ P ∗
λj T yj ∈ oE . So we have T x = j yj (T x) yj = j λj xj (x) yj .

3.57 Corollary (See [Kri07a, 5.28], [Jar81, 19.8.6 p.442]).

We have A1 ⊆ N1 .

Proof. See [Jar81, P19.8.5 p.442]. Let T ∈ A1 (E, F ). We have to show that it can
be written as T = n λn x∗n ⊗ yn with x∗n ∈ oE ∗ , yn ∈ oF and λ ∈ `1 .
Let ε > 0. Choose Tn with dim Tn (E) ≤ 2n and kT − Tn k ≤ (1 + ε) a2n (T ). Let
Dn := Tn+1 − Tn . Then dn := dim Dn (E) ≤ 3 · 2n and since an (T ) → 0 we have
P∞ P∞ Pdn
kT − Tn k → 0, hence T = n=0 Dn . By 3.56 we have T = n=0 j=1 λn,j x∗n,j ⊗
yn,j , with x∗n,j ∈ oE ∗ , yn,j ∈ oF and 0 ≤ λn,j ≤ kDn k. We estimate as follows
dn
XX X X
λn,j ≤ dn kDn k ≤ 3 2n (kTn+1 − T k + kTn − T k)
n j=1 n n
X
2n (1 + ε) a2n+1 (T ) + a2n (T )


≤3·
n
X X
≤3· 2n+1 (1 + ε) a2n (T ) ≤ 22 3 (1 + ε) 2n−1 a2n (T )
n n
X
2
≤ 2 3 (1 + ε) an (T ) (since an (T ) is decreasing)
n

to conclude that (λn,j )n,j ∈ `1 .

3.58 Lemma (See [MV92, 28.14 p.334], [Jar81, 21.6.1 p.496]).

Diagonal operators on `p (for 1 ≤ p < ∞) are nuclear iff they have `1 coefficients.
Cf. 3.10 and 3.11 .
P
Proof. (⇐) obvious, since D = dn evn ⊗en with ken k`p = 1 = k evn k`q
n
p ∗
(⇒) Let a ∈ ` = (` ) , b ∈ ` with n ka(n) k`q · kb(n) k`p < ∞ and D(x) =
(n) q (n) p
P
P (n) P (n) (n)
na (x) b(n) for all x ∈ `p . With x := ek we get dk = D(ek )k = n ak · bk
(n) (n)
and hence kdk`1 = k |dk | ≤ k,n |ak | · |bk | ≤ n ka(n) k`q · kb(n) k`p < ∞ by
P P P
the Hölder-inequality.
We will apply this to the connecting mappings ιkn : λk → λn for the Köthe-sequence
spaces λ = λp (A) with 1 ≤ p ≤ ∞. Only the case p = ∞ needs special attention
(see 1.13 ): Let the diagonal operator D := ιkn : λk → λn be nuclear. Then
D|c0 : c0 ,→ λk → λn ,→ `∞ is nuclear, so a(n) ∈ `1 = (c0 )∗ and b(n) ∈ `∞ and
hence the same proof as above for p = 1 shows that the diagonal d of D is absolutely
summable.

3.59 Proposition. Factorization property of N

(See [Kri07a, 5.6 p.119], [Jar81, 17.3.2 p.377]).

42 andreas.kriegl@univie.ac.at c July 1, 2016

Operator ideals 3.61

A map T : E → F between Banach spaces is nuclear iff there are continuous linear
operators S : E → `∞ and R : `1 → F such that T factors as diagonal operator
D : `∞ → `1 with diagonal d ∈ `1 , i.e.

E
T /F
O
S R

`∞ / `1
D

Proof. (⇒) Let T be represented by k dk x∗k ⊗yk P with kx∗k kE ∗ ≤ 1, kyk kF ≤ 1 and
P
1 ∗
d ∈ ` . Then S(x) := (xk (x))k and R((µk )k ) := k µk yk define linear operators
S : E → `∞ and R : `1 → F of norm ≤ 1 and T = R ◦ D ◦ S, where D : `∞ → `1
denotes the diagonal operator, with diagonal (dk )k .
(⇐) Since the nuclear operators form an ideal, it is enough to show that such
∞ 1
diagonal operators D : (µk )k 7→ (d
Pk µk )k ,∗D : ` → ` are nuclear, which is clear
since they can be represented by k dk xk ⊗ yk , where x∗k := ek ∈ `1 ⊆ (`∞ )∗ and
yk := ek ∈ `1 .

3.60 Lemma. N ⊆ K (See [Kri07a, 5.7], [Jar81, 17.3.4 p.379]).

Every nuclear operator is compact.

Proof. Let T be a nuclear mapping. Since the compact mappings form an ideal,
we may assume by 3.59 that T is a diagonal-operator `∞ → `1 with absolutely
summable
P diagonal (λk )k . Such an operator is compact, since the finite sub-sums
k≤n λ k e k ⊗ ek define finite dimensional operators, which converge to T uniformly
on the unit-ball of `∞ .

3.61 Lemma (See [Kri07a, 5.19], [Jar81, 19.5.1 p.428]).

Every p-summing operator induces a continuous linear map from `p [E] → `p {F }.
Thus we may consider the space Sp (E, F ) of p-summing operators as normed sub-
space of the space L(`p [E], `p {F }).

Here we consider the space `p {F } supplied with the norm

X 1/p
k(yk )k kπ := kyk kpF .
k

p
As in 3.42 one can show that ` {F } is complete (see [Jar81, 19.4.1 p.426]). For
p > 1 it is however not isomorphic to `p ⊗ ˆ π F . Otherwise we would obtain for
ˆ π `p = `p {`p } = `p (N × N), which is not the case..
E = `p , that `p ⊗
On `p [E] we consider the operator norm of L(`q , E) (see 3.43 ):
nX 1/p o
k(xk )k kε := sup |x∗ (xk )|p : x∗ ∈ E ∗ , kx∗ k ≤ 1 .
k

It is obvious, that the inclusion `p {E} → `p [E] is a contraction (i.e. has norm ≤ 1).

Proof. Let T : E → F be a p-summing operator. We will apply the closed graph

theorem to T∗ : `p [E] → `p {E}, (xn )∞ ∞
n=1 7→ (T (xn ))n=1 , so consider x
(k)
→ x in
p (k) p
` [E] with T∗ (x ) → y in ` {F }. Since obviously kT∗ (z)kε ≤ kT k · kzkε with
respect to the operator norms k kε , we get ky − T∗ (x)kε ≤ ky − T∗ (x(k) )kε +
kT∗ (x(k) − x)kε ≤ ky − T∗ (x(k) )kπ + kT k kx(k) − xkε → 0, and hence T∗ (x) = y.

andreas.kriegl@univie.ac.at c July 1, 2016 43

3.64 Operator ideals

3.62 Corollary (See [Kri07a, 5.20], [Jar81, 19.5.2 p.428]).

An operator T : E → F is p-summing iff there exists a R > 0 such that
X 1/p X 1/p
k(T x(k) )k kπ := kT x(k) kp ≤ R· sup |x∗ (x(k) )|p =: R·k(x(k) )k kε .
k kx∗ k≤1 k

for all finite sequences (x )k . The smallest such R is the norm of T∗ : `p [E] →
(k)

`p {F }, and is also denoted kT kSp . Consequently, N1 ⊆ S1 .

Proof.
(⇒) By 3.61 we have that T∗ is continuous, and hence the required property holds
with R := kT∗ k and all (even the infinite) sequences in `p [E].
(k) p 1/p
Pm

(⇐) For x = (x(k) )k ∈ `p [E] we have k(T x(k) )k kπ = supm k=1 kT x k ≤
R · k(x(k) )k≤m kε ≤ R · k(x(k) )k kε < ∞ and hence (T x(k) )k ∈ `p {F }.
(N1 ⊆ S1 ) by 3.59 , since any diagonal operator D : `∞ → `1 with diagonal d ∈ `1
is 1-summing:
m
(k) (k)
X XX X X X
kDx(k) k`1 = |dj xj | = |dj | |xj | ≤ kdk`1 sup | evj (x(k) )|
j
k=1 k j j k k
nX o
≤ kdk`1 sup |x∗ (x(k) )| : x∗ ∈ (`∞ )∗ , kx∗ k ≤ 1 .
k

3.63 Proposition (See [Kri07a, 5.21], [Jar81, 19.5.4 p.430]).

For p ≤ q we have Sp ⊆ Sq .

Also Np ⊆ Nq can be shown under the same assumption, see [Jar81, 19.7.5 p.437].
1 1 1
Proof. Let T ∈ Sp and let r ≥ 0 be given by r + q = p. Let λk := kT xk kq/r .
r/q
Then kT xk k = λk and hence kT (λk xk )kp = kλk T (xk )kp = λpk · kT xk kp =
q
kT xk kp( r +1) = kT xk kq and so the Hölder’s inequality (cf. the proof of 3.53 )
shows that
X 1/p X 1/p
kT xk kq = kT (λk xk )kp = (T (λk xk ))k π
k k
X 1/p
≤ kT kSp · (λk xk )k ε
= kT kSp · sup |x∗ (λpk xk )|p
kx∗ k≤1 k
X 1/r X 1/q
∗
≤ kT kSp · λrk · sup |x (xk )|q
k kx∗ k≤1 k
X 1/r X 1/q
≤ kT kSp · kT xk kq · sup |x∗ (xk )|q
k kx∗ k≤1 k
P 1/r P 1/p−1/q
Dividing by k kT xk kq = k kT xk kq gives
X 1/q  1/q
kT xk kq ≤ kT kSp · sup |x∗ (xk )|q .
k kx∗ k≤1

Thus T ∈ Sq by 3.62 .

3.64 Lemma. Summing via measures

(See [Kri07a, 5.22], [Jar81, 19.6.1 p.431]).

44 andreas.kriegl@univie.ac.at c July 1, 2016

Operator ideals 3.65

An operator T is p-summing iff there exists some probability measure µ on the

compact unit ball oE ∗ and an M > 0 such that
Z 1/p
kT xk ≤ M · |x∗ (x)|p dµ(x∗ ) .
oE ∗

Proof. Note that the right hand side is nothing else but M · kδ(x)kp , where
δ : E  C(oE ∗ ).
(⇐) If µ is a probability measure (i.e. µ(oE ∗ ) = 1) with that property, then
X Z X nX o
kT xk kp ≤ M p |x∗ (xk )|p dµ(x∗ ) ≤ M p · sup |x∗ (xk )|p : x∗ ∈ oE ∗ .
k oE ∗ k k

So T ∈ Sp by 3.62 .
(⇒) Let T ∈ Sp (E, F ). For every finite sequence x = (x1 , . . . , xm ) in E let fx ∈
C(oE ∗ ) be defined by
X X X 
fx (x∗ ) := kT kpSp · |x∗ (xi )|p − kT xi kp = kT kpSp · |x∗ (xi )|p − kT xi kp .
i i i
(N)
The set B := {fx : x ∈ E } is convex in C(oE ∗ ). In fact let x and y be two
finite sequences in E and λ + µ = 1 with λ ≥ 0 and µ ≥ 0. Let z be the sequence
obtained by appending µ1/p y to λ1/p x. Then
X  
(λfx + µfy )(x∗ ) = λ kT kpSp |x∗ (xi )|p − kT xi kp +
i
X  
+ µ kT kpSp |x∗ (yj )|p − kT yj kp
j
X
= (kT kpSp )|x∗ (λ1/p xi )|p − kT (λ1/p xi )kp +
i
X
+ (kT kpSp )|x∗ (µ1/p yj )|p − kT (µ1/p yj )kp
j
X
= (kT kpSp )|x∗ (zk )|p − kT (zk )kp = fz (x∗ ).
k

By 3.62 we have that supx∗ ∈oE ∗ fx (x∗ ) ≥ 0. Thus the open set A := {f ∈
C(oE ∗ ) : supx∗ ∈oE ∗ f (x) < 0} is disjoint from B. So by the consequence [Kri07b,
7.2.1] of Hahn-Banach there exists a regular Borel measure µ on oE ∗ and a constant
α such that hµ, f i < α ≤ hµ, gi for all f ∈ A and g ∈ B. Since 0 ∈ B we have α ≤ 0.
Since A contains the constant negative functions we have α = 0 and µ(oE ∗ ) > 0.
Without loss of generality we may assume kµk = 1. Hence for every x ∈ E we have
Z  
0 ≤ hµ, fx i = kT kpSp |x∗ (x)|p − kT xkp dµ(x∗ )
oE ∗
p
and thus kT xk ≤ kT kSp · oE ∗ |x∗ (x)|p dµ(x∗ ).
p
R

3.65 Theorem. Factorization of absolutely 2-summing operators

(See [Kri07a, 5.24], [Jar81, 19.6.4 p.433]).
The operators T in S2 are characterized by the existence of a compact space K and
a measure µ on K such that we have the following factorization:

E
T /F
O


 i / L2 (µ)
C(K)

andreas.kriegl@univie.ac.at c July 1, 2016 45

3.67 Operator ideals

Proof. (⇐) It is enough to show that the canonical mapping ι : C(K) → L2 (µ) is
absolutely 2-summing. So let δx be the point measure at x. Then for finitely many
fk ∈ C(K) we have
X Z X Z X
kι(fk )k2`2 = |fk (x)|2 dµ(x) = |δx (fk )|2 dµ(x)
k K k K k
nX o
≤ µ(K) · sup |ν(fk )| : ν ∈ C(K)∗ , kνk ≤ 1 ,
2

hence the natural mapping ι belongs to S2 by 3.62 .

(⇒) By 3.64 there is some probability measure µ ∈ M(oE ∗ ) such that

Z 1/2
kT xk ≤ M · |x∗ (x)|2 dµ(x∗ ) .
oE ∗

The map δ : E → C(oE ∗ ), x 7→ evx , is isometric. Now consider the diagram

E
T /F
_ < O
R
S ## R
H q /H
δ
1 O
P
 "
C(oE ∗ )
ι / L2 (µ)

where H denotes the closure of the image of ι◦δ in L2 (µ). The operator T factorizes
via a continuous linear operator R : H → F , since kT xk ≤ M · kι(δ(x))k`2 for some
M > 0. Using the ortho-projection P : L2 (µ) → H we get the factorization
R ◦ P ◦ (ι ◦ δ) = R ◦ ι ◦ δ = R ◦ S = T .

3.66 Proposition (See [Kri07a, 5.31], [Jar81, 20.5.1 p.467]).

For Hilbert spaces we have S2 ⊆ A2 .

Proof.
For orthonormal families ek and fk and T ∈ S2 we have by 3.61
X 1/2 X 1/2
kT ek k2 ≤ kT kS2 · sup |hx, ek i|2 ≤ kT kS2 .
k kxk≤1 k

And by the Cauchy-Schwarz inequality |hT ek , fk i| ≤ kT ek k · kfk k = kT ek k we get

2 2 2
P P
k |hT ek , fk i| ≤ k kT ek k < kT kS2 < ∞, hence T ∈ A2 by 3.54 .

3.67 Overview. One has the following inclusions for 1 < p < q < ∞:

 3.57
/ N 1  
3.62
/ S1
A 1 _ _
_
obvious [Jar81, 19.7.5 p.437] 3.63
   
N p 
[Jar81, 19.7.8 p.438]
A p / Sp
_ _  _
obvious [Jar81, 19.7.5 p.437] 3.63
   
Nq 
[Jar81, 19.7.8 p.438]
Aq / Sq

46 andreas.kriegl@univie.ac.at c July 1, 2016

Operator ideals 3.69

For Hilbert spaces one has the following results for 1 < p < ∞:


N 1 
[Jar81, 20.2.5 p.456]
A1 / S1
_ _
[Jar81, 20.5.1 p.467]
 [Jar81, 20.5.1 p.467]  [Jar81, 20.5.1 p.467]
A2 Np Sp
_ _
[Jar81, 20.5.1 p.467]
  
A∞ N∞  / S∞

Nuclear spaces

In this section we characterize nuclear spaces in several ways and we prove their
inheritance properties We show that the nuclear (Fréchet) spaces are exactly the
(closed) subspaces of products of (countable many) copies of s.

3.68 Definition.
A linear mapping T : E → F between lcs is called nuclear operator (See [Jar81,
17.3 p.376], [Kri07a, 5.6]) iff there exist {an : n ∈ N} ⊆ E ∗ equicontinuous, B a
Banach disk, bn ∈ B, and λ ∈ `1 with

∞
X
Tx = λn an (x) bn for all x ∈ E.
n=1

This is exactly the case, iff there is an absolutely convex 0-neighborhood U ⊆ E and
a Banach disk B ⊆ F , such that T factors over a nuclear mapping T̃ : E cU → FB ,
i.e.

E
T /F
O


E
fU / FB
T̃

The nuclear mappings form an ideal: For composition from the left side with some
R replace bn by R(bn ), and from the right side replace an by an ◦ R = R∗ (an ) (Note
that an ∈ U o ⇒ R∗ (an ) ∈ (R−1 (U ))o ).

3.69 Proposition (See [Jar81, 17.3.8 p.380]).

Let T : E → F be nuclear and G any lcs. Then T ⊗ G : E ⊗ε G → F ⊗π G is
continuous.

Note, that as for any bifunctor we denote with T ⊗ G the morphism T ⊗ idG .

o
P
Proof. We may represent T = n λn an ⊗ bn with an ∈ U for some 0-nbhd.
U and bn ∈ B, a Banach-disk. Let V ⊆ F and W ⊆ G be 0-nbhds and let

andreas.kriegl@univie.ac.at c July 1, 2016 47

3.70 Nuclear spaces

Pk
ρ := sup{qV (b) : b ∈ B}. For w = j=1 xj ⊗ zj ∈ E ⊗ G we get
k
X X
(T ⊗ G)(w) = T (xj ) ⊗ zj = λn an (xj ) bn ⊗ zj
j=1 j,n
X X
= λ n bn ⊗ an (xj ) zj and hence
n j
  ∞
X k
X 
πV,W (T ⊗ G)(w) = πV,W λ n bn ⊗ an (xj )zj
n=1 j=1
∞
X k
nX o
≤ |λn | qV (bn ) sup an (xj )z ∗ (zj ) : z ∗ ∈ W o
n=1 j=1
∞
X n k
X o
≤ρ |λn | sup x∗ (xj )z ∗ (zj ) : x∗ ∈ U o , z ∗ ∈ W o
n=1 j=1
≤ ρ kλk`1 εU,W (w).

3.70 Theorem. Characterizing nuclear spaces in multiple ways

(See [Kri07a, 6.17], [Jar81, 21.2.1 p.482]).
Let 1 ≤ p < ∞. Then
1.
E is nuclear;
⇔ E ⊗π F = E ⊗ε F for every Banach space F ;
2.
⇔ E ⊗π `1 = E ⊗ε `1 ;
3.
⇔ `1 {E} = `1 hEi topologically;
4.
⇔ `1 {E} = `1 [E] topologically;
5.
⇔ 6.
The connecting maps of the projective representation can be chosen abso-
lutely summing (or Sp );
⇔ 7. The connecting maps of the projective representation can be chosen nuclear
(or Np );
⇔ 8. The connecting maps of the projective representation can be chosen 1-approximable
(or Ap );
⇔ 9. Every continuous linear map into a Banach space is nuclear.

Proof. We give the proof for 1 ≤ p ≤ 2 only. For the general case one needs in
addition that Sp ◦ Sq ⊆ Sr (see [Jar81, 19.10.3 p.446]) and N1 ⊆ Np ⊆ Sp (see
[Jar81, 19.7.5 p.437] and [Jar81, 19.7.8 p.438]).
( 1 ⇒ 2 ⇒ 3 ) and ( 5 ⇒ 4 ⇒ 3 ) are obvious by 3.42 and 3.50 .
( 3 ⇒ 6 ) From ( 3 ) we obtain that `1 hẼi ∼
= `1 {Ẽ}. Thus for every U ⊆ E there
exists a V ⊆ E and a δ > 0 such that πU ≤ δ εV , where

X
πU (xk )k := pU (xk )
k

is the semi-norm associated to U on ` ⊗π E ∼1ˆ

= `1 {E}, see 3.42 , and where
nX o
εV ((xk )k ) := sup |y ∗ (xk )| : y ∗ ∈ V o
k

is the semi-norm associated to U on L(c0 , F ) = `1 [E] and hence on the subspace

`1 hEi ∼
= `1 ⊗
ˆ ε E, see 3.50 . From this it follows by 3.62 that the connecting map
is absolutely summing.

48 andreas.kriegl@univie.ac.at c July 1, 2016

Nuclear spaces 3.72

( 6 ⇒ 5 ) For every U we can find by assumption a V such that the connecting

map ιVU : EV → EU is absolutely summing. Hence if (xk )k ∈ `1 [E], then the images
are in `1 [ẼV ] and hence in `1 {ẼU }. Moreover, by 3.62
n
X n
X
ιVU (ιV (xk ))


πU (xk )k := pU (xk ) = U
≤
k=1 k=1
nX o
≤ kιVU kS1 · sup |x∗ (xk )| : x∗ ∈ V o =: kιVU kS1 · εV ((xk )k ),
k

Since U was arbitrary we have ( 5 ).

( 7 ⇒ 1 ) By assumption for every U there exists a U 0 such that the connecting
0
U0
map ιUU is nuclear. By 3.69 we have that ιU ⊗ F̃V : ẼU 0 ⊗ε F̃V → ẼU ⊗π F̃V is
continuous. Thus πU,V ≤ c · εU 0 ,V for some c > 0, i.e. E ⊗ε F = E ⊗π F . Recall the
corresponding norms on EU ⊗π EV and on EU ⊗ε EV :
nX X o
πU,V (z) := inf pU (xk ) pV (yk ) : z = xk ⊗ yk and
k k
X  nX o
εU 0 ,V xk ⊗ yk := sup x∗ (xk ) y ∗ (yk ) : x∗ ∈ (U 0 )o , y ∗ ∈ V o
k k

( 6 ⇔ 7 ⇔ 8 ) Now let us show that for all mentioned ideals it is the same to
assume that the connecting mappings belong to them.
In fact, we have A1 ⊆ N1 ⊆ S1 ⊆ Sp ⊆ S2 for 1 ≤ p ≤ 2 by 3.57 , 3.62 , 3.63 .
The composite of three S2 maps belongs to A2 , since the following diagram shows
that it factors over a map between Hilbert spaces (see 3.65 ) of class S2 ⊆ A2 (by
3.66 ):

E3
S2
/ E2 S2
/ E1 S2
/ E0
; ;

# S2 ⊆A2
#
L2 (µ3 ) / L2 (µ1 )

Since (A2 )2 ⊆ A1 by 3.53 we have that (S2 )6 ⊆ A1 . Now choose for a given
seminorm p successively p6 ≥ p5 ≥ · · · ≥ p1 ≥ p such that the connecting maps all
belong to S2 . Then the connecting mapping E gp6 → Ep belongs to A1 .
f

( 7 ⇔ 9 ) Recall that a map T : E → F with values in a Banach T /F

E O
space is called nuclear (see 3.68 and 3.59 ), iff it factors over
a nuclear map T1 : E1 → F on some Banach space E1 . In fact, ιV T̃
 ιU
!
for E1 we may choose E fU for some 0-neighborhood U . Now we
/ ẼU
ẼV
can proceed as for the corresponding result 3.30 for compact ιV
U

mappings and Schwartz spaces.

3.71 Characterizing nuclear (F) spaces via summable sequences

(See [Kri07a, 6.18], [Jar81, 21.2.4 p.483]).
A Fréchet space is nuclear iff `1 {E} = `1 [E] (or `1 {E} = `1 hEi) holds algebraically.

Proof. Since `1 {E} and `1 hEi are Fréchet spaces it follows from the closed graph
theorem that the identity is a homeomorphism.

3.72 Corollary. Nuclear spaces have a basis of Hilbert seminorms

(See [MV92, 28.1 p.325], [Kri07a, 6.19.1]).

andreas.kriegl@univie.ac.at c July 1, 2016 49

3.73 Nuclear spaces

Proof. By what we have shown in the proof of ( 6 ⇒ 8 ) in 3.70 using 3.65

every natural mapping E → Ep factors over some Hilbert-space H. Taking the
norm q of the Hilbert-space, we get a continuous seminorm E → H −q→ R, which
dominates p.

3.73 Inheritance properties for nuclear and Schwartz spaces

(See [Kri07a, 6.21], [Jar81, 21.2.3 p.500], [Jar81, 21.1.7 p.481]).
Both the nuclear and the Schwartz spaces are stable with respect to:
1. products, (See [MV92, 28.7 p.328], [Kri07a, 6.21], [Jar81, 21.1.3 p.479],
[Jar81, 21.1.4 p.480])
2. subspaces, (See [MV92, 28.6 p.328], [Kri07a, 6.21], [Jar81, 21.1.5 p.481],
[Jar81, 21.1.6 p.481])
3. countable coproducts, (See [MV92, 28.7 p.328], [Kri07a, 6.21], [Jar81,
21.1.3 p.479], [Jar81, 21.1.4 p.480])
4. quotients, (See [MV92, 28.6 p.328], [Kri07a, 6.21], [Jar81, 21.1.5 p.481],
[Jar81, 21.1.6 p.481])
5. completions, (See [Jar81, 21.1.2 p.481])
6. projective tensor products, (See [Jar81, 15.6.5 p.337])

Proof.
Q
( 1 ) A typical seminorm on E := Ei is of the form p : x 7→ maxi∈A pi (xi ), where
i
A is finite and pi are seminorms on Ei . Obviously Efp = Q ^
i∈A (Ei )pi . For every pi
we can find a seminorm qi ≥ pi such that the canonical mapping (^ ^
Ei )qi → (Ei )pi

is precompact/nuclear. Then the canonical Q

Q ^ Q ^
mapping i∈A (Ei )qi → Pi∈A (Ei )pi is
precompact/nuclear, in fact a finite product i∈A Ti can be written as i∈A inji ◦Ti ◦
pri and hence belongs to the considered ideal. Thus we may use q := maxi∈A qi as
the required seminorm.
( 2 ) First for Schwartz spaces. Let E be a subspace of F . The seminorms on E are
the restrictions of seminorms p on F . Let q ≥ p be a seminorm such that Fq → Fp
is precompact. Since Ep|E → Fp is an embedding (ker p|E = ker p ∩ E) we have the
diagram:
Eq|E / Ep|
_ _ E

 
Fq / Fp

Since the bottom arrow is precompact, the same is true for the top arrow.
Now for nuclear spaces. The corresponding proof will not work for nuclear map-
pings, but for absolutely summing mappings, since the ideal S1 is obviously injec-
tive, i.e. if T : E → F1 ,→ F belongs to S1 and F1 is a closed subspace of F , then
T : E → F1 belongs to S1 , since `1 {F1 } = `1 {F } ∩ F1N .
( 3 ) First for`
Schwartz spaces. Recall that a basis of seminorms on a countable co-
product E = k Ek is given by supk pk , where the pk run through the seminorms of
Ek and supk pk : (xk )k 7→ supk pk (xk ). By assumption we can find seminorms qk ≥
pk such that the connecting map Tk : (Ek )qk → (Ek )pk is precompact. Furthermore
we may assume that its norm is less than 21k , by replacing qk with k
` 2 kTk kq∼
k.
Now the following diagram shows that we get a natural bijection k (Ek )pk =

50 andreas.kriegl@univie.ac.at c July 1, 2016

Nuclear spaces 3.73

` `
( k Ek )supk pk which is an isometry if we supply k (Ek )pk with the norm (xk )k 7→
sup{pk (xk ) : k}, and analogously for the qk .
`
ker(supk pk ) k ker(p )
_ _ k
`  ` 
k Ek k Ek
supk pk w
` pk
'
Ro Rg
sup k
7
`
k

 
∼
( k Ek )supk pk o /
` = `
k (E k )pk

Up to these isometries the connecting map is nothing else but

a a a
T := Tk : (Ek )qk → (Ek )pk .
k k k
` `
Since the finite subsums k≤n Tk converge uniformly to k Tk on the unit-ball with
respect to p = supk pk and are precompact operators by the result on products,
hence so is the infinite sum.
Now for nuclear spaces. We proceed P as before using that the connecting mappings
Tk can be chosen of the form Tk = j (λk )j (x∗k )j ⊗(yk )j with λk ∈ `1 and sequences
x∗k ∈ o((Ek )qk )o and yk ∈ o((Ek )pk ). By replacing qk by kλk k1 2k qk , we have that
1
P1 , λ2 , . . . ) ∗∈ ` and hence the connecting
(λ
∗
mapping T admits the representation
k,j (λ )
k j (x )
k j ⊗ (y )
k j , where (x )
k j can be extended to the corresponding space,
since (Ek )qk embeds isometrically into it.
( 4 ) First for Schwartz spaces. Let F := E/N , where N is a closed subspace and
let π : E → F denote the quotient mapping. Let p̃ be a seminorm on F . By
assumption there exists a seminorm q on E with q ≥ p̃ ◦ π and such that Eq → Fp̃◦π
is precompact. Let q̃ be the corresponding quotient semi-norm on F , see [Kri07b,
4.3.3]. Then q ≥ q̃ ◦ π ≥ p̃ ◦ π. Now the following diagram shows that we get a
natural isometry Ep̃◦π ∼ = E p̃ and similarly for q̃.

ker π  / ker(p̃ ◦ π) π −1 (ker p̃) / / ker p̃
_
_

  
N  /E //F
π

p̃◦π p̃

'
7 R xe
p̃◦π p̃
 
∼
Ep̃◦π / / / Fp̃
=

Another argumentation for the same result would be an application of the isomorphy-
theorem F/ ker p̃ ∼
= (E/N )/(ker(p̃ ◦ π)/N ) ∼
= E/ ker(p̃ ◦ π).
Hence we have the diagram: Note that connecting morphisms are always quotient
Fq̃ / Fp̃ maps, since the projections E → Eq are. So the di-
== O O agonal arrow is open, since it is up to the vertical
∼
= ∼
= isomorphism the connecting map Eq → Eq̃◦π . Hence
the image of the unit ball in Eq is a 0-nbhd in Fq̃
/ / Eq̃◦π / / Ep̃◦π .
Eq whose image is precompact in Ep̃◦π ∼
= Fp̃ .

andreas.kriegl@univie.ac.at c July 1, 2016 51

3.74 Nuclear spaces

Now in order that this proof works also for nuclear spaces, we can use the following:
It is enough to consider the situation, where E  E1 → F is nuclear, E  E1 is a
quotient map and E a Hilbert space (by 3.72 ). But then the sequence E2  E 
E1 splits, where E2 is the kernel of the quotient map E  E1 , and hence E1 → F
can be written as E1 ,→ E  E1 → F and thus is nuclear.

( 5 ) Use that E fq = E f̃ ,
q̃ ker
_ q E ∩ ker q̃  / ker q̃
_
where q̃ denotes the unique   
extension of q to a seminorm E / / Ẽ
on Ẽ.
) u
6Rh
q q̃


q q̃

  / / Ẽq̃
Eq

( 6 ) First for Schwartz spaces. Recall that the typical 0-neighborhoods of E ⊗π F

are the absolutely convex hulls U1 ⊗ U2 of {u1 ⊗ u2 : u1 ∈ U1 , u2 ∈ U2 }, where
the Ui are absolutely convex 0-neighborhoods in Ei . By assumption there are 0-
neighborhoods Vi ⊆ Ui in Ei such that for every 0 < ε ≤ 1 there is a finite set Bi
such that Vi ⊆ Bi + ε Ui . Taking intersection with Ui shows that Vi ⊆ (Bi + ε Ui ) ∩
Ui ⊆ (Bi ∩ 2Ui + ε Ui ). In fact b + ε u ∈ Ui implies that b ∈ Ui − ε u ⊆ Ui − Ui ⊆ 2Ui .
Thus we may assume that Bi ⊆ 2Ui . Now we have that
V1 ⊗ V2 ⊆ B1 ⊗ B2 + ε B1 ⊗ U2 + ε U1 ⊗ B2 + ε2 U1 ⊗ U2
⊆ B1 ⊗ B2 + (2ε + 2ε + ε2 ) U1 ⊗ U2 .
So let V := 51 V1 ⊗ V2 ⊆ 4+ε1
V1 ⊗ V2 and B := 4+ε 1
B1 ⊗ B2 . Then V ⊆ B + ε U .
Since B is the absolutely convex hull of a finite set, it is precompact, hence we can
find a finite set B0 such that B ⊆ B0 + ε U , and so V ⊆ B0 + 2εU .
For nuclear spaces E and F we take an arbitrary lcs G and calculate as follows:
E nucl.
(E ⊗π F ) ⊗ε G ∼
= (E ⊗ε F ) ⊗ε G ∼
= E ⊗ε (F ⊗ε G)
F nucl. E nucl.
∼
= E ⊗ε (F ⊗π G) ∼
= E ⊗π (F ⊗π G) ∼
= (E ⊗π F ) ⊗π G.

3.74 Nuclearity of λp (A) (See [MV92, 28.16 p.335]).

Let A = {a(k) : k ∈ N} be countable. Then
1. ∃p ∈ [1, ∞]: λp (A) (N);
⇔ 2. ∀p ∈ [1, ∞]: λp (A) (N);
⇔ 3. c0 (A) (N);
⇔ 4. ∃p, q : 1 ≤ p < q ≤ ∞ und λp (A) = λq (A);
⇔ 5. ∀p, q : 1 ≤ p < q ≤ ∞ ⇒ λp (A) = λq (A);
⇔ 6. ∀k ∃m ≥ k: ka(k) /a(m) k1 < ∞.

Proof. ( 2 ⇒ 3 ) λ∞ (A) (N) ⇒ λ∞ (A) is (S) by 3.60 and hence (M) by 3.31
⇒ λ∞ (A) = c0 (A) by 3.28 .
( 1 ⇒ 6 ) and ( 3 ⇒ 6 ) follows for p < ∞ from 3.58 for the diagonal operators
D∼ = ιm p ∞
k on ` resp. c0 , hence also for λ (A) = c0 (A).

( 6 ⇒ 2 ) follows from 3.58 for p < ∞. By 3.59 the diagonal operator `∞ → `1

with diagonal d = a(k) /a(m) ∈ `1 is nuclear and hence its composite λm ,→ `∞ →
`1 → `∞ is nuclear and thus absolutely summming, and so also the connecting
homomorphism λm → λk is absolutely summing.

52 andreas.kriegl@univie.ac.at c July 1, 2016

Nuclear spaces 3.77

( 2 ⇒ 1 ) obvious.
( 4 ⇔ 5 ⇔ 6 ) follows from 1.25

One can show the following:

3.75 Theorem of Dynin-Mityagin

(See [MV92, 28.12 p.332], [Jar81, 21.10.1 p.510]).
For nuclear Fréchet spaces ((NF) for short) each Schauder-basis is absolute (recall
1.17 ).

3.76 Corollary (See [MV92, 28.13 p.334]).

Each (NF) space with Schauder-basis (ej )j is isomorphic to λ1 (A), where
A := {j 7→ kej kk : k ∈ N}.

Proof. This is a direct consequence of 3.75 and 1.21 .

It was open for a long time whether all (NF) spaces have a Schauder-basis. The
first counter-example was given in [MZ74], see [Jar81, 21.10.9 p.516] for a sim-
pler counter-example. Rather recently it was shown in [DV00] that the complete
ultra-bornological nuclear space C ω (R, R) of real-analytic functions does not have
a Schauder-basis.

3.77 Theorem of Grothendieck-Pietsch

(See [MV92, 28.15 p.334] ([Jar81, 21.6.2 p.497])).
A nuclearity criterium for (F) with Schauder-basis (ej )j∈N is:
X kej kk
∀k ∃m ≥ k : < ∞.
j
kej km

Proof. (⇒) Let E be (NF) with a Schauder-basis (ej )j . Then E ∼ = λ1 ({a(k) :

k ∈ N}) with aj := kej kk by 3.76 and thus the claimed condition is satisfied by
(k)

3.74 .
P p(e )
(⇐) For any continuous seminorm p choose p0 with j p0 (ejj ) < ∞. By 1.19 there
exists a p00 and C > 0 such that
∀x ∀j : |ξj (x)| p0 (ej ) ≤ C p00 (x),
where ξj are the coefficient functionals. Then ξj factors (for p0 (ej ) 6= 0) over
ιp00 : E  Ep00 to a ξ˜j ∈ (Ep00 )∗ . Thus D : Ep00 → Ep , x 7→ j=0 ξ˜j (x) ιp (ej ), is a
P∞
nuclear mapping, since
∞ ∞ ∞
X X X p(ej )
kξ˜j k kιp (ej )k = sup{|ξj (x)| : p00 (x) ≤ 1} p(ej ) ≤ C < ∞.
j=0 j=0 j=0
p0 (ej )
00
Thus the connecting mapping ιpp is nuclear, since it equals D:
∞
X ∞
X
˜


(D ◦ ι )(x) =
p00 ξj ιp (x) ιp (ej ) =
00 ξj (x) ιp (ej )
j=0 j=0
∞
X  00
= ιp ξj (x) ej = (ιpp ◦ ιp00 )(x).
j=0

andreas.kriegl@univie.ac.at c July 1, 2016 53

3.79 Nuclear spaces

3.78 Nuclearity of power series spaces λR (α)

(See [MV92, 29.6 p.344], [Jar81, 21.6.3 p.497]).

log(n)
1. λ∞ (α) is nuclear ⇔ supn αn < ∞.
2. λ0 (α) is nuclear ⇔ limn log(n)
αn = 0.

3.74
e−tαn < ∞
P
Proof. ( 1 ) λ∞ (α) nuclear ⇐ ⇒ ∃t > 0 : C :=
==== n
n
−tαn
X log(n) log(C) log(C)
(⇒) ⇒ ne ≤ e−tαj ≤ C ⇒ ≤ +t≤ + t =: D.
j=1
αn αn α0
log(n) 1 X
(⇐) sup ≤ D ⇒ e−Dαn ≤ ⇒ e−2Dαn < ∞
n αn n n

(2)
3.74 X
λ0 (α) is nuclear ⇐ ⇒ ∀t > 0 :
==== e−tαn < ∞, now proceed as in ( 1 ).
n

Example (See [Jar81, 21.6.4 p.498]).

s = λ∞ (log(n)) is nuclear and hence also the function spaces in 1.16 ;
λ0 (log(n)) is not nuclear, but Schwartz by 3.35 .

3.79 Lemma (See [MV92, 29.7 p.344]).

Let E be (N), p a continuous Hilbert SN, and U := {x : p(x) ≤ 1} its unit-ball.
Then there exists a fast-falling ONB (en )n∈N of EU∗ o , i.e.
∀k ∃V : {nk en : n ∈ N} ⊆ V o .

Proof.
3.70.8 , 3.53
⇒ ∀k > 0 ∃pk ≥ p, cont. Hilbert SN : ιppk ∈ A1/k (Epk , Ep )
E (N ) ===========
As in 3.33 : (Ep )∗ ∼
= EU∗ o , (Epk )∗ ∼
= EU∗ ko with Uk := {x : pk (x) ≤ 1}

⇒ ιk := (ιppk )∗ : EU∗ o  EU∗ ko , ιk ∈ A1/k (EU∗ o , EU∗ ko )

3.54
====
=
X (k) (k) (k)
y = ιk (y) = aj hy, ej ifj with
j
(k) (k)
(ej )j ON in EU∗ o , (fj )j ON in EU∗ ko ,
(k)
X (k)
(aj )j ↓, C 1/k := (aj )1/k < ∞.
j
m
(k)
X
⇒ m(a(k)
m )
1/k
≤ (aj )1/k ≤ C 1/k , d.h. a(k)
m ≤ C/m
k

j=1
(k)
und (ej )j ONB in EU∗ o , da ιk inj.
(j)
Let (ẽn )n be the diagonal-enumeration of (ei )i,j , drop recursively those which
are linearly dependent on ealier ones, and apply Gram-Schmidt to obtain an ONB

54 andreas.kriegl@univie.ac.at c July 1, 2016

Nuclear spaces 3.82

(en )n in EU∗ o . Let n ≥ (2k)2 .

√
√ X 1 n+ n
∀j < n−k : i= (j + k)(j + k + 1) < ≤n
2 2
i≤j+k
√ (k)
⇒ ∀j < n − k : en ⊥ ej , since then ẽn lies on a diagonal below (j, k)
√ √
√  n n (k)
⇒ jn := n−k ≥ (since k ≤ ) and ∀j < jn : en ⊥ ej
2 2
∞ ∞
(k) (k) (k) (k) (k)
X X
⇒ hen , ej i ej = en = ιk (en ) = aj hen , ej i fj
j=jn j=jn
∞ ∞
(k) (k) (k) (k)
X X
⇒ ken k2Uko = |aj hen , ej i|2 ≤ |ajn |2 |hen , ej i|2 ≤
j=jn j=jn

C2 2 (2k C)2 Ck
≤ 2k
ke n k U o ≤ k
=: k for all large n
(jn ) n n
⇒ (en )n fast falling in E ∗ .

3.80 Theorem of Komura-Komura

(See [MV92, 29.8 p.346], [Jar81, 21.7.1 p.500]).
Let E be an lcs: E is (N) ⇔ ∃I: E ,→ sI := i∈I s.
Q

Proof. (⇒)
E (N ) ⇒ ∃(pi )i∈I basis of Hilbert SN, let Ui := {x : pi (x) ≤ 1}
3.79
⇒ ∀i ∃(ein )n fast falling ONB in EU∗ io
====
=
⇒ ∀k ∃Vk : {nk ein : n} ⊆ (Vk )o , i.e. ∀x ∈ Vk : sup |nk ein (x)| ≤ 1
n
⇒ fi : E → s, x 7→ (ein (x))n , is continuous
I
⇒ f := (fi )i∈I : E → s is continuous.
∀x ∈ E : evx = διi (x) is continuous on the Hilbert space EU∗ io = (EUi )∗
[Kri07b, 6.2.9]
⇒ ∃x∗ ∈ EU∗ io ∀y ∗ ∈ EU∗ io : hx∗ , y ∗ i = evx (y ∗ ) = y ∗ (x)
============
X X
⇒ kfi (x)k20 := |ein (x)|2 = |hx∗ , ein i|2 = kx∗ k2E ∗ o = k evx k2 = pi (x)2
U
i
n n
I
⇒ f is an embedding onto f (E) ⊆ s .

3.73.1 , 3.73.2
(⇐) s (N), E ,→ sI = ⇒ E (N).
============

3.81 Nuclear Fréchet spaces (See [MV92, 29.9 p.346], [Jar81, 21.7.3 p.502]).
E is (NF) ⇔ E is isomorphic to a closed linear subspace of sN .

Proof. (⇐) sN is (NF) by 3.78 and 3.73.1 , thus also E by 3.73.2 .

(⇒) For the (NF) space E exists a countable basis P of Hilbert SN and by the
proof of 3.80 E embedds into sP .

3.82 Remark.
Note that we have the following implications under the assumption on the bottom

andreas.kriegl@univie.ac.at c July 1, 2016 55

3.82 Nuclear spaces

of the arrow:

3.60 3.31 3.22 3.17

⇒ Schwartz =
====
nuclear = ⇒ s.-Montel =
==== ⇒ s.-reflexive =
==== ⇒ q.-complete.
====
q.-compl.

The converse does not hold even for Fréchet spaces:

3.78 3.36 `2 c0 , `1
nuclear 6⇐= = Schwartz 6⇐=
==== = Montel 6⇐=
==== = reflexive 6⇐==== complete.
=

56 andreas.kriegl@univie.ac.at c July 1, 2016

 4. Duality

Spaces of (linear) functions

In this section we discuss how the hom-functor behaves on (co-)limits.

Let X be a set and B be a bornology on X, i.e. a set of subsets of X containing all

single pointed subsets and with any two sets a set containing their union. For lcs F
let `∞ (X, F ) be the linear space of all mappings f : X → F , which are bounded on
each B ∈ B. For continuous seminorms p of F and B ∈ B let pB be the seminorm
on `∞ (X, F ) definied by pB (f ) := sup p(f (B)). These seminorms describe the
Hausdorff topology of uniform convergence on the sets B ∈ B. Obviously, `∞ (B, F )
is as complete as F is for each B ∈ B and hence the same is true for the projective
limit `∞ (X, F ) ∼
= lim `∞ (B, F ), see [Kri07a, 2.28]. Note that the lcs `∞ (X, F )
←−B∈B
will not change, iff we add all subsets of sets in B to B.
The space L(E, F ) of bounded linear mappings E → F between lcs (or even from
a convex bornological space into an lcs) is closed in `∞ (E, F ), where B are the
bounded subsets of E, and hence has the same completeness properties as F . In
particular, E 0 is always complete with respect to β(E 0 , E). Note that a convex
bornological space (cbs for short) is a linear space together with a bornology,
which is closed under formation of absolutely convex hulls and multiplication with
(say) 2. We will always assume that cbs are separated, i.e. {0} is the only bounded
linear subspace. The von Neumann bornology of all bounded sets of an lcs E
describes a cbs b E and conversely to any cbs F we may associate the finest locally
convex topology t F for which the sets in the given bornology are bounded, i.e.
with the corresponding bornivorous absolutely convex subsets a 0-nbhd basis. See
[Gac04] for more on this concept.
For the space L(E, F ) of continuous linear mappings and for the particular case E ∗
these completeness inheritance properties are not valid. However, if E is bornolog-
ical then L(E, F ) = L(E, F ) and E ∗ = E 0 .
More generally the question arrises, whether L( , F ) (or in particular ( )∗ ) trans-
forms inductive limits into projective ones. By the universal property algebraically
the dual of a colimit is the limit of the duals: The continuous linear mappings
on a colimit E := colimj Ej correspond uniquely to the families of morphisms
fj : Ej → F with (ιjj 0 )∗ (fj ) = fj 0 ◦ ιjj 0 = fj for all j ≺ j 0 , i.e. which are
compatible with respect to the connecting morphisms ιjj 0 : Ej → Ej 0 . These
are the elements in the limit of the L(Ej , F ) with connecting mappings (ιjj 0 )∗ =
L(ιjj 0 , F ) : L(Ej 0 , F ) → L(Ej , F ). However, this linear (continuous) bijection
L(colimj Ej , F ) → limj L(Ej , F ) is not to be expected an homeomorphism, since
for a typical 0-nbhd. B o in E ∗ = L(E, K) with QB ⊆ E bounded, we would have
to find 0-nbhds. Bjo in Ej∗ with B o ⊇ E ∗ ∩ j∈J Bjo and such that Bjo = Ej∗
for almost all j. This is possible, if E is a regular inductive limit (i.e. colimj Ej

andreas.kriegl@univie.ac.at c July 1, 2016 57

4.2 Spaces of (linear) functions

formed in cbs), since then B = ιj (Bj ) with Bj ⊆ Ej bounded for some j and hence
B o ⊇ ((ιj )∗ )−1 (Bjo ), but not in general.
Note, that the representation E = limB EB of a bornological space is such a regular
−→
inductive limit.
Note furthermore, that if ( )∗ is supplied with the bornology of equicontinuous sets,
then (colimj Ej )∗ = limj Ej∗ in cbs: In fact, let U o be a typical bounded set in E ∗ ,
i.e. U ⊆ E := colim Ej a 0-nbhd.QThen each Uj := (ιj )−1 (U ) ⊆ Ej is Q a 0-nbhd
and the image of U o in limj Ej∗ ⊆ j Ej∗ is contained in the bounded set j (Uj )o ,
since |(ιj )∗ (u∗ )(uj )| = |u∗ (ιj (uj ))| ≤ 1 for u∗ ∈ U and uj ∈ Uj .
We want to consider inheritance with respect to L or L. If E 6= {0}, then F is
a topological direct summand in L(E, F ) and in L(E, F ): In fact, let 0 6= x ∈ E
and x∗ ∈ E ∗ with x∗ (x) = 1. Then ι : K → E, λ 7→ λ · x has x∗ : E → K as
= L(K, F ) → L(E, F ) has L(ι, F ) : L(E, F ) →
left-inverse, and hence L(x∗ , F ) : F ∼
L(K, F ) ∼= F as left-inverse and the same works for L. And similary, F ∗ = L(F, K)
is a topological direct summand in L(F, E), via L(F, ι) with left-inverse L(F, x∗ )
and the same way F 0 = L(F, K) is a topological direct summand in L(F, E). Thus
in order to show some topological property for L(E, F ) it is reasonable to assume
the property for F and for E ∗ . Consequently a first step in answering this question
is to consider inheritance with respect to ( )∗ .

Completeness of dual spaces

In this section we consider completeness conditions for the (strong) dual and we
introduce the classes of infra-c0 -barrelled and of c0 -barrelled space in this connec-
tion.

Recall the Banach Steinhaus Theorem [Kri07b, 5.2.6], by which L(E, F ) is sequen-
tially complete if E is barrelled and F is sequentially complete:
Let (fn )n be a Cauchy-sequence in L(E, F ). Then (fn )n∈N is Cauchy pointwise,
hence pointwise convergent to some function f∞ : E → F , which is continuous by
the Banach Steinhaus Theorem. For each bounded B ⊆ E and closed absolutely
convex 0-nbhd U ⊆ F there exists an n with (fn0 − fn00 )(B) ⊆ U for n0 , n00 ≥ n.
Taking for each x ∈ B the pointwise limit for n00 → ∞ yields (fn0 − f∞ )(x) ∈ U .
Thus fn → f∞ in L(E, F ).

4.1 Example of a non-complete dual space.

Let F be a barrelled non-complete space (in [Val71] even normed bornological bar-
relled spaces are constructed, which are not ultra-bornological and hence not even
locally complete). Let E := (F ∗ , σ(F ∗ , F )). Thus F = E ∗ and by the barrelledness
of F the σ(F ∗ , F )-bounded subsets are the equicontinuous ones. Hence the topol-
ogy β(E ∗ , E) coincides with the topology of uniform convergence on equicontinuous
sets and hence with the given non-complete topology of F .
If E is infra-barrelled, then the dual E ∗ is at least locally complete: In fact, under
this assumption the β(E ∗ , E)-bounded sets are equicontinuous and EU∗ o = (EU )∗
is complete as dual of a normed space. In order to improve this result, we need the
following characterization:

4.2 Proposition (See [Jar81, 10.2.4 p.198], [Woz13, 2.39 p.21]).

For any lcs E we have:
1. E is locally complete;
⇔ 2. The absolutely convex hull of every Mackey-0-sequence is relatively compact;

58 andreas.kriegl@univie.ac.at c July 1, 2016

Completeness of dual spaces 4.4

⇔ 3. The absolutely convex hull of every σ(E, E ∗ )-0-sequence is relatively compact

in (E, σ(E, E ∗ )).

Proof. ( 1 ⇒ 3 ) Let xn → 0 in σ(E, E ∗ ). Then {xn : n ∈ N} is (weakly-)bounded

and by locally
P∞ completeness bounded in some closed Banach disk B. Thus T : `1 →
1
E, λ 7→ n=0 λn xn , is well-defined and maps the unit ball o` onto
nX ∞ o
A0 := λn xn : kλk`1 ≤ 1 ⊆ B.
n=0

It is σ(`1 , c0 )-σ(E, E ∗ )-continuous, since x∗ ◦ T = (x∗ (xn ))n∈N ∈ c0 ⊆ (`1 )∗ for each
x∗ ∈ E ∗ . Since o`1 is σ(`1 , c0 )-compact, its image A0 is σ(E, E ∗ )-compact, abso-
lute convex, and contains the xn . Hence their absolutely convex hull is relatively
compact for σ(E, E ∗ ).
( 3 ⇒ 2 ) Let xn be a Mackey-0-sequence. By 3 the σ(E, E ∗ )-closure C of the
absolutely convex hull of {xn : n ∈ N} is σ(E, E ∗ )-compact and hence σ(E, E ∗ )-
complete. Since closed absolutely convex sets in E are σ(E, E ∗ )-closed, C is even
complete in E by the next lemma 4.3 . Since {xn : n ∈ N} ∪ {0} is compact, its
closed absolutely convex hull is precompact (by the proof of [Kri07b, 6.4.3]) and
thus compact by completeness of C.
( 2 ⇒ 1 ) Suppose there is a closed absolutely convex bounded set B, such that EB
is not complete. Choose x̃ ∈ E fB \ EB and iteratively construct a sequence (xi )i∈N
in EB such that
n
X 1
x̃ − xi ≤ n+2
i=1
B 3
P∞
and hence x̃ = i=1 xi . Now let yn := 2n xn ∈ EB and observe that
 Xn n−1
X   2 n+1
n
kyn kB ≤ 2 x̃ − xi + x̃ − xi ≤ → 0.
i=1
B
i=1
B 3
P∞ −n
Hence x̃ = n=1 2 yn is in the closure of the absolutely convex hull of the
(Mackey-)0-sequence (yn ) in the Banach space E fB . Consider the initial topol-
ogy τ with respect to the inclusion ι : EB  E. Since B is closed in E, it is closed
0

for τ 0 , thus (EB , k kB ) has a basis of τ 0 -closed sets. By the lemma 4.4 below the
extension ι̃ : E fB  Ẽ is injective. Since ι̃(x̃) = P∞ 2−n yn is in the (by 2 )
n=1
compact closure of the absolutely convex hull of {yn : n ∈ N} ⊆ B in E, we get
x̃ ∈ E ∩ B ⊆ EB , a contradiction.

4.3 Lemma (See [Jar81, 3.2.4 p.59]).

Let τ ≥ τ 0 be two lc-topologies on a vector space E and assume that (E, τ ) has a
0-nbhd basis U consisting of τ 0 -closed subsets.
If (xi )i is τ -Cauchy net in E, which converges to x∞ with respect to τ 0 , then it does
so with respect to τ .
Thus, if a subset of E is (sequentially) complete for τ 0 , then it is also for τ .

Proof. Cf. the proof of the corollary to 3.17 : Let (xi )i be a τ -Cauchy net,
which is τ 0 -convergent to x∞ , and let U ∈ U. Thus there exists an i such that
xi0 − xi00 ∈ U for all i0 , i00  i. For fixed i0 , the net i00 7→ xi0 − xi00 ∈ U is τ -Cauchy
and τ 0 -convergent to xi0 − x∞ . Since U is τ 0 -closed we get xi0 − x∞ ∈ U , i.e. (xi )i
is τ -convergent to x∞ .

4.4 Lemma (See [Jar81, 3.4.5 p.63]).

Let (E, τ ) be an lcs, T ∈ L(E, F ) be injective and τ 0 ≤ τ be the initial topology

andreas.kriegl@univie.ac.at c July 1, 2016 59

 Completeness of dual spaces

on E with respect to T . If (E, τ ) has a 0-nbhd-basis of τ 0 -closed subsets then the

extension T̃ : Ẽ → F̃ to the completions is injective.

Proof. Let x̃ ∈ ker T̃ ⊆ Ẽ. Thus there exists a net (xi )i in E convergent to x̃ in
Ẽ and hence T (xi ) = T̃ (xi ) → T̃ (x̃) = 0. Thus xi → 0 with respect to τ 0 and then
x̃ = τ - limi→∞ xi = 0 by 4.3 .

4.5 Proposition (See [Jar81, 12.1.4 p.250], [Woz13, 2.64 p.30]).

The dual E ∗ of an lcs E is locally complete iff (E, µ(E, E ∗ )) is infra-c0 -barrelled,
i.e. every 0-sequence in (E ∗ , β(E ∗ , E)) is equicontinuous.
Thus for infra-c0 -barrelled spaces the dual is barrelled iff it is infra-barrelled, and it
is ultra-bornological iff it is bornological.
Furthermore, an lcs E is called c0 -barrelled iff every 0-sequence in (E ∗ , σ(E ∗ , E))
is equicontinuous, see [Jar81, 12.1 p.249].
Proof. (⇒) Let x∗n → 0 in (E ∗ , β(E ∗ , E)) and hence in (E ∗ , σ(E ∗ , E)). So their
closed absolutely convex hull K is σ(E ∗ , E)-compact by 4.2 ( 1 ⇒ 3 ). Thus Ko
is a 0-nbhd of µ(E, E ∗ ) and x∗n ∈ K ⊆ (Ko )o .
(⇐) By 4.2 ( 2 ⇒ 1 ) it is enough to show that for any Mackey-0-sequence (x∗n )
in E ∗ its absolutely convex hull A is relatively compact w.r.t. β(E ∗ , E). Any such
sequence is equicontinuous (by the infra-c0 -barrelledness), hence A is relatively
compact for σ(E ∗ , E) and thus the closure of A is complete. Since β(E ∗ , E) has
a 0-nbhd basis of σ(E ∗ , E)-closed sets (B o ), it is also β(E ∗ , E)-complete by 4.3 .
Since {x∗n : n ∈ N} ∪ {0} is β(E ∗ , E)-compact, we get that the closed absolutely
convex hull of the sequence is precompact and hence compact w.r.t β(E ∗ , E)..

4.6 Proposition (See [Jar81, 11.2.4 p.222]).

E infra-barrelled ⇒ E ∗ is quasi-complete.

Proof. Let B ⊆ Eβ∗ be bounded. Since E is infra-barrelled, B is equicontinuous, i.e.

B ⊆ U o for some 0-neighborhood. The polar U o is σ(E ∗ , E)-compact by 3.4 , hence
σ(E ∗ , E)-complete and therefore also β(E ∗ , E)-complete by 4.3 , since β(E ∗ , E)
has a basis of σ(E ∗ , E)-closed subsets (B o ).

Barrelledness and bornologicity of dual spaces

In this section we give conditions that garantee barrelledness or ultra-bornologicity

of the strong dual. For this we show that the (appropriate) duality functor preserves
reduced projective limits and products. We introduce the classes of (infra-)countably-
barrelled spaces and discuss their relationship to the other barrelledness conditions.

If order to show that E ∗ is bornological, we have to represent E ∗ as inductive

limit of normed spaces. So it is reasonable to assume that E is representable as
projective limit of normed spaces. Because of E ∗ = Ẽ ∗ (at least bornologically)
it is no big restriction to assume that E is complete and hence E = limU ẼU , a
←−
reduced projective limit of Banach spaces. Remains to check, whether
?
E ∗ = (lim ẼU )∗ = lim(EU )∗ .
←− −→
U U

60 andreas.kriegl@univie.ac.at c July 1, 2016

Barrelledness and bornologicity of dual spaces 4.9

For any functor F we have a natural morphism F(lim Xi ) → lim F(Xi ) by the
universal property of the right side.
F (Xα )
XT i
Xα
/ Xj F(Xi ) / F(Xj )
J T a = J
pri prj

limk F(Xk )
pri prj F (pri ) O F (prj )
!

limk Xk F(limk Xk )

4.7 Lemma. Reduced projective limits (See [Kri07a, 3.25]).

Let limi Xi be a reduced projective / / Yi
←− XO i e e
O fi : O
limit, and fi : Xi → Yi be con-
tinuous linear mappings with dense pri lim f pri
−i i/ / lim Y
limi Xi ←
image which intertwine with all con- ←− ←−i i
pri0 pri0
necting mappings. Then the canoni-
cal mapping limi fi has dense image. yy fi0 $
←− Xi0 / / Yi0

Proof. Let z ∈ limi Yi be given. Take an arbitrary 0-neighborhood pr−1 i (2 Ui ).

←−
Since fi has dense image we may find an xi ∈ Xi with fi (xi ) − pri (z) ∈ Ui . Since
the first limit is reduced we can find an x ∈ E with pri (x) − xi ∈ fi−1 (Ui ). But
then
 
pri lim fi (x) − z = (fi ◦ pri )(x) − fi (xi ) + fi (xi ) − pri (z) ∈ 2Ui ,
←−
i
i.e. limi fi has dense image.
←−
4.8 Lemma. The dual of products (See [Kri07a, 3.26]).
The functor ( )∗ : lcs → cbsop preserves products, where E ∗ is considered with the
bornology of equicontinuous sets.
Here cbs denotes the category of convex bornological spaces with those linear map-
pings, which map bounded sets to bounded sets, as morphisms.
Proof.`By the general argument above we have a mapping i Ei∗ → ( i Ei )∗ ,
` Q
where ∗
i Ei denotes the coproduct `in cbs and hence the product in cbsop . Since
∗
Q
i Ei obviously separates points in i Ei this mapping is injective. Let us show
that it is a bornological quotient map, i.e. bounded sets in the image are im-
ages
Q of bounded sets. This implies that it is a bornological Q isomorphism. So let
( i Ui )o be a typical bounded:=equicontinuous subset of ( i Ei )∗ , i.e. the Ui are
0-neighborhoods of Ei and Ui = Ei for all i ∈/ J, where J is a finite subset of I.
Let T ∈ ( i Ui )o . Then T (x) = 0 for all x =
Q
Q(xi )i with xj = 0 for all j ∈oJ (use∗
that every multiple P x belongs to `i Ui ). Let Ti := T ◦ inj`
of such an ` i ∈ Ui ⊆ Ei
for all i. Then T = j∈J Tj ∈ j∈J Ujo and j∈J Ujo is bounded in i Ei∗ .

4.9 Lemma. The dual of reduced projective limits (See [Kri07a, 3.27]).
The functor ( )∗ : lcs → cbsop preserves reduced projective limits, where E ∗ is again
considered with the bornology of equicontinuous sets.

Proof. So let E := limi Ei be a reduced projective limit. As in the proof of 4.8

←−
we have a natural mapping limi Ei∗ → (lim Ei )∗ . Since all projections pri : E  Ei
−→ ←−

andreas.kriegl@univie.ac.at c July 1, 2016 61

4.11 Barrelledness and bornologicity of dual spaces

have dense image the dual cone pr∗i : Ei∗  E ∗ consists of injective mappings only.
Let x∗ ∈ E ∗ be given. Then there has to exist an i and a 0-neighborhood Ui ⊆ Ei
with x∗ (pr−1i (Ui )) ⊆ D := {λ ∈ K : |λ| ≤ 1}. In particular x (ker pri |E ) = 0 and
∗

hence there exists a linear xi : pri (E) → R with x = xi ◦ pri = pr∗i (x∗i ). Since
∗ ∗ ∗

i (Ui )) ⊆ D we may extend xi to a continuous functional

x∗i (Ui ∩ pri (E)) = x∗ (pr−1 ∗

in Ui ⊆ (Ei ) on the closure Ei of pri (E). Thus the union of all images pr∗i ((Ei )∗ )
o ∗

is E ∗ . Moreover the same argument shows that every bounded:=equicontinuous set

(pr−1 o o ∗
i (Ui )) is the image of the bounded set Ui under pri . From this it is clear that
∗
(limi Ei ) is the injective limit in cbs, since any family of bounded linear mappings
←−
Ti : Ei∗ → F that commute with the S connecting morphisms can be extended to a
bounded linear mapping T : E ∗ = i pr∗i (Ei∗ ) → F .
fU is a reduced projective limit, so (Ẽ)∗ = lim (EU )∗ =
In particular, Ẽ = limU E
∗
←− −→U
limU EU o as convex bornological space (with respect to the equicontinuous bornol-
−→
ogy). But this does not imply that it is true for the strong topology and this
topology on (Ẽ)∗ need not be bornological.
What about infra-barrelledness of E ∗ ?
Let V ⊆ E ∗ be a bornivorous barrel, so V absorbs every bounded set in E ∗ and,
in particular, the polars U o of (closed absolutely convex) 0-nbhds U in E. From
K · V ⊇ U o we conclude, that Vo ⊆ K · (U o )o = K · U , i.e. Vo is bounded, and
thus (Vo )o is a 0-neighborhood, but not necessarily (contained in) V , since β(E ∗ , E)
need not be compatible with duality (E ∗ , E).

4.10 Proposition (See [Tre67, p373], [Kri07a, 4.47]).

The strong dual of any semi-reflexive space is barrelled.

An lcs E is sometimes called distinguished iff E ∗ barrelled, see [Jar81, 13.4.5

p.280].

Proof. Let V be a barrel in Eβ∗ . Since E is semi-reflexive the strong topology

is compatible with the duality, and hence V is also closed for the weak-topology
σ(E ∗ , E) by [Kri07b, 7.4.8]. We show that the polar Vo is a bounded subset of E
(which implies that V = (Vo )o is a 0-neighborhood in Eβ∗ ). For this it is enough to
show that Vo is bounded in σ(E, E ∗ ): Since V is assumed to be absorbing, we find
for every x∗ ∈ E ∗ a λ > 0 with x∗ ∈ λ V . Thus |x∗ (Vo )| ≤ λ.

4.11 Proposition (See [MV92, 24.23 p.267], [Woz13, 3.52 p.56]).

The strong dual of any complete Schwartz space is ultra-bornological.

Proof. Let E be a complete Schwartz space. By 3.31 it is semi-Montel, hence

β(E ∗ , E) = τc (E ∗ , E) = τpc (E ∗ , E), by completeness. By the theorem 3.4 of
Alaoǧlu-Bourbaki U o is σ(E ∗ , E)-compact (and even τpc (E ∗ , E)-compact) for all
0-nbhds U and therefore by [Kri07b, 7.4.17] is a Banach disk. The inclusions ιU o :
EU∗ o  (E ∗ , τc (E ∗ , E)) are bounded=continuous and therefore η ≥ τc (E ∗ , E) ≥
σ(E ∗ , E), where η denotes the ultra-bornological final locally convex topology on
E ∗ generated by these mappings.
To see the converse τc (E ∗ , E) ≥ η, we choose 0-nbhds V ⊆ U such that U o is
compact in EV∗ o (by 3.33 ). By continuity of ιV o : EV∗ o  (E ∗ , η) the polar
U o is compact in (E ∗ , η) and therefore id : (U o , η) → (U o , σ(E ∗ , E)) is a homeo-
morphism, i.e. σ(E ∗ , E) = η on U o , and, since γ(E ∗ , E) is the finest such locally
convex topology, γ(E ∗ , E) ≥ η and γ(E ∗ , E) = τc (E ∗ , Ẽ) = τc (E ∗ , E) by 3.24
and completeness.

62 andreas.kriegl@univie.ac.at c July 1, 2016

Barrelledness and bornologicity of dual spaces 4.14

4.12 Definition (See [Jar81, 12.2 p.251]).

An lcs E is called (infra-)countably-barrelled (or (quasi-)ℵ0 -barrelled)
iff every countable intersection of closed absolutely convex 0-nbhds is a 0-nbhd
provided it is (bornivorous) absorbing.
By the following proposition we get:
(infra-)barrelled ⇒ (infra-)countably-barrelled ⇒ (infra-)c0 -barrelled,

4.13 Proposition (See [Jar81, 12.2.1 p.252]).

Let E be an lcs. Then
1. E is (infra-)countably-barrelled;
⇔ 2. For any lcs F every (β)σ-bounded sequence in L(E, F ) is equicontinuous;
⇔ 3. For any Banach space F every (β)σ-bounded sequence in L(E, F ) is equicon-
tinuous.
Here β denotes the topology of uniform convergence on each bounded set and σ that
of pointwise convergence.

Proof. ( 1 ⇒ 2 ) Let Tn ∈ L(E, F ) be a sequence as considered in 2 . Let V be

a closed absolutely convex 0-nbhd
S in F . For every finite (resp. bounded) set B ⊆ E
there exists a ρ > 0 such that n∈N Tn (B) ⊆ ρV . Thus U := n∈N Tn−1 (V ) is an
T

absorbing (resp. bornivorous) absolutely convex set, hence a 0-nbhd by 1 . Since

Tn (U ) ⊆ V for all n, we get that {Tn : n ∈ N} is equicontinuous.
( 2 ⇒ 3 ) is trivial.
T
( 3 ⇒ 1 ) Let U = n∈N Un be absorbing (resp. bornivorous) with Un absolutely
N
Q 0-nbhds. Let F := {x ∈ E : x is finally constant}. The subset
convex closed
V := F ∩ n∈N Un is absolutely convex and absorbing (since U absorbs the finite
set {xj : j ∈ N} for x ∈ F ) in F . Let Tn : E → F be T given by x 7→ (xi )i∈N
with xi := x for i ≤ n and xi := 0 for i > n. Then Tn ( i≤n Ui ) ⊆ V and hence
T̃n = ιV ◦Tn : E → F  FfV is continuous. Since U is absorbing (resp. bornivorous),
the set {T̃n : n ∈ N} is σ(resp. β)-bounded (B ⊆ λU ⇒ Tn (B) ⊆ λV ), hence
equicontinuous by 3 , so there exists a 0-nbhd W ⊆ E with 2 T̃n (WT) ⊆ ιV (V ) ∩
FV = ιV (V ) for all n. Thus ∀w ∈ W ∃v ∈ V : 2 Tn (w)−v ∈ ker pV = λ>0 λV ⊆ V
and, in particular, 2 w = 2(prn (Tn (w))) ∈ prn (v) + prn (V ) ⊆ 2Un , i.e. W ⊆ Un for
all n ∈ N, hence W ⊆ U and we are done.

4.14 Lemma (See [Jar81, 12.2.2 p.252]).

Every locally complete infra-countably-barrelled lcs is countably-barrelled.
Every locally complete infra-c0 -barrelled lcs is c0 -barrelled.
T
Proof. Let V = n∈N Vn be absorbing as required in the definition 4.12 . So V is a
barrel, hence absorbs Banach-disks by the Banach-Mackey-Theorem (See [Kri07b,
7.4.18]). Since in locally complete lcs every closed bounded absolutely convex
set is a Banach-disk, V is even bornivorous, hence a 0-nbhd by infra-countably-
barrelledness.
The same proof works for (infra-)c0 -barrelledness with Vn := {x∗n }o for a given
0-sequence x∗n in Eσ∗ , cf. the proof of 1 ⇒ 2 in 4.13 .

Remark.
We have shown the following implications, where the dotted ones are valid under

andreas.kriegl@univie.ac.at c July 1, 2016 63

4.16 Barrelledness and bornologicity of dual spaces

the assumption of c∞ -completeness:

ulta-bornological
go
 '/
barrelled bornological
go
 '/ 
countably-barrelled infra-barrelled
go
 '/ 
c0 -barrelled infra-countably-barrelled
go
'/ 
infra-c0 -barrelled

Duals of Fréchet spaces

In this section we describe the property (DF), which the strong duals of Fréchet
spaces have, and which garantees in turn that their strong dual is Fréchet.

4.15 Lemma (See [MV92, 25.6 p.279], [Jar81, 12.2.4 p.253]).

Let E be metrizable. Then E ∗ is countably-barrelled.
o ∗
n ⊆ E is bounded. Let Vn be
Proof. Let (Un )n be a 0-nbhd-basis of E. Then UT
∗
closed absolutely convex 0-nbhds in E and V∞ := n∈N Vn be bornivorous.
Recursively we will find ρi > 0 and Bi ⊆ E bounded such that
1
Bio ⊆ Vi and ρi Uio ⊆ i+2 V∞ ∩ Bjo for all i, j ≤ n.
2
For (n = 0) take a bounded set B0 ⊆ E such that B0o ⊆ V0 and find aT ρ0 > 0
with ρ0 U0o ⊆ 41P 1
V∞ ∩ B0o . For the induction step choose ρn Uno ⊆ 2n+2 V∞ ∩ i<n Bio .
o ∗
The set K := i≤n ρi Ui is absolutely convex, σ(E , E)-compact, and contained in
1 1 0 1 ∗
P
i≤n 2i+2 V∞ ⊆ 2 Vn . Let V ⊆ 2 Vn be a σ(E , E)-closed absolutely convex 0-nbhd
in E . Then Bn := (V + K)o is bounded and Bno = V 0 + K ⊆ Vn by the bipolar
∗ 0

theorem.
Thus W := n Bno ⊆ E ∗ satisfies W = (Wo )o and absorbs each Uio , hence Wo is
T
bounded and thus (Wo )o = W ⊆ V is a 0-nbhd. in E ∗ . This shows infra-countably-
barrelledness. Since E ∗ is complete, countably-barrelledness follows by 4.14 .

4.16 Proposition (See [MV92, 25.12 p.281], [Jar81, 13,4, p.280]).

Let E be a metrizable lcs. Then
1. E∗ is ultra-bornological;
⇔ 2. E∗ is bornological;
⇔ 3. E∗ is barrelled;
⇔ 4. E∗ is infra-barrelled.

We will give an example (of a non-distinguished λ1 (A)) in 4.25 for which these
equivalent conditions are not satisfied.
In [Jar81, 13.4.2 p.279] it is shown that for metrizable E the bornologification
β(E ∗ , E)born of β(E ∗ , E) is β(E ∗ , E ∗∗ ).

64 andreas.kriegl@univie.ac.at c July 1, 2016

Duals of Fréchet spaces 4.19

Proof. ( 1 ⇒ 2 ⇒ 4 ) are obvious.

( 4 ⇒ 3 ) and ( 2 ⇒ 1 ) since E ∗ = E 0 is complete.
( 3 ⇒ 2 ) Let V be absolutely convex and bornivorous in E ∗ . Thus for every
bounded=equicontinuous set Uno (where the Un from a 0-nbhd basis S in E) there
exists a λn > 0 with V ⊇ 2λn Uno . Let U be the absolutely convex hull of n∈N λn Uno .
Then U ⊆ 12 V . The absolutely convex hull Ak of j≤n λj Ujo is σ(E ∗ , E)-compact
S

(by exercise 34 to [Kri07a]), hence closed in β(E ∗ , E) ≥ σ(E ∗ , E).

We claim that Ū ⊆ V : Let x∗0 ∈ E ∗ \ V ⊆ E ∗ \ 2U . Since AnTis closed there exists
a 0-nbhd Vn ⊆ E ∗ with (x∗0 + Vn ) ∩ 2An = ∅. Let W := n∈N (Vn + An ). Let
k ∈ N. Then Uko ⊆ λ1k An for all n ≥ k. Choose µk ≥ 1/λk with Uko ⊆ µk Vn for
all n < k. Thus Uko ⊆ µk (Vn + An ) for all n, i.e. Uko ⊆ µk W , i.e. W is bornivorous
and hence a 0-nbhd in E ∗ by 4.15 . We claim that (x∗0 + W ) ∩ An = ∅ for all n
and hence x∗0 ∈ / Ū , since otherwise ∅ 6= (x∗0 + W ) ∩ An ⊆ (x∗0 + Vn + An ) ∩ An , i.e.
∃v ∈ Vn , ∃a, a0 ∈ An : a = x∗0 + v + a0 . Hence x∗0 + v = a − a0 ∈ 2An and thus
(x∗0 + Vn ) ∩ 2An 6= ∅, a contradiction.
So the barrel Ū ⊆ V . Since E ∗ is assumed to be barrelled, we are done.

4.17 Definition. (DF)-spaces (See [Jar81, 12.4.1 p.257], [MV92, 25.6 p.279]).
An lcs E is called (DF)-space, iff it has a countable base of the bounded sets
and is infra-countably-barrelled (see 4.12 ), i.e. every bornivorous subsets which is
the intersection of countable many closed absolutely convex 0-neighborhoods is a
0-neighborhood.
An lcs E is called (df)-space iff it has a countable base of its bornology and is
infra-c0 -barrelled.

4.18 Proposition.

1. The dual of any Fréchet space is a complete (DF) space

(See [MV92, 25.7 p.280], [Jar81, 12.4.5 p.260]).
2. The dual of any (DF) space is a Fréchet space
(See [MV92, 25.9 p.280], [Jar81, 12.4.1 p.257]).
In [Jar81, 12.4.1 p.257] it is shown that: E ∗ is Fréchet ⇔ (E, µ(E, E ∗ )) is (df).

Proof. ( 1 ) This is 4.15 , since for the bornological space E the dual E ∗ = E 0 is
complete, and a countable basis of the bornology is given by the family Uno , where
{Un : n ∈ N} is a 0-nbhd basis of E.
( 2 ) By assumption a (DF)-space E has a countable base {Bn : n ∈ N} of bornology
and hence (E ∗ , β(E ∗ , E)) a countable 0-nbhd basis {Bno : n ∈ N}, so is metrizable.
Let (x∗n )n be Cauchy in E ∗ . ThenT x∗n converges pointwise to some linear x∗∞ : E →
K. Let Vn := {xn }o and V∞ := n∈N Vn . Since (x∗n )n is Cauchy, it is bounded,
∗

thus contained in λk Bko for some λk > 0. Hence Bk ⊆ λk {x∗n }o = λk Vn and so

Bk ⊆ λk V∞ , i.e. V∞ is bornivorous and hence a 0-nbhd since E is infra-countably-
barrelled. Furthermore, x∗n ∈ (Vn )o ⊆ (V∞ )o , hence x∗∞ ∈ (V∞ )o ⊆ E ∗ . And since
(by 4.3 ) the Cauchy-sequence x∗n converges to x∗∞ uniformly on Bk for any k ∈ N,
we get that x∗n → x∗∞ in E ∗ .

4.19 Corollary (See [MV92, 25.10 p.51]).

The bidual of any Fréchet space is a Fréchet space.

andreas.kriegl@univie.ac.at c July 1, 2016 65

4.21 Duals of Fréchet spaces

Duals of Köthe sequence spaces

In this section we describe the duals of Köthe sequence spaces and characterize
reflexivity (and the Montel property) of λ∞ (A) (and of c0 (A)). We also give an
example of a Köthe sequence space, whose strong dual fails to be (infra-)barrelled.

4.20 Seminorms of λp (A)∗ (See [MV92, 27.13 p.314]).

Let A be countable, λ := λp (A) for 1 ≤ p < ∞ or λ := c0 (A) for p = ∞. Then the
Minkowski-functionals (Bbp )o for Bbp := {x : kx/bk`p ≤ 1} (see 2.10 ) are given by
1 1
kykb := ky · bk`q , for b ∈ λ∞ (A) and + = 1,
p q
and are a basis of the seminorms of
= y ∈ KN : ∀b ∈ λ∞ (A) : kykb < ∞ = y ∈ KN : ∃a ∈ A : kyk∗a < ∞ ,
n o n o
λ∗ ∼
o
where k k∗a is the Minkowski-functional of x ∈ λ : kxka ≤ 1 (see 1.24 ).


P∞
Proof. By 1.22 we have λ∗ ∼


= λ1 (λ) via x 7→ j=0 xj yj ← y.
By 2.10 the sets Bbp := {x : kx/bk`p ≤ 1} (resp. Bbo : Bb∞ ∩ c0 (A)) for b ∈ λ∞ (A)
(w.l.o.g. ∀j : bj > 0) form a basis of the bornology on λp (A) (resp. c0 (A)). Let
y ∈ λ1 (λ) ∼
= λ∗ and 1q + p1 := 1, then by 1.23 the Minkowski-functional p(Bbp )o is
given by
∞ nXx o
j
X
sup |y(x)| = sup xj yj = sup bj yj : kx/bk`p ≤ 1 = ky · bk`q =: kykb .
x∈Bbp x∈Bbp j=0 j
bj

= y ∈ KN : ∀b ∈ λ∞ (A) : kykb < ∞ , since y ∈ KN acts as bounded(=continuous)

λ∗ ∼


linear functional ⇔ ∀b ∈ λ∞ (A) : kykb < ∞.

λ∗ ∼= y ∈ KN : ∃a ∈ A : kyk∗ < ∞ , since λ∗ = λ∗ o , where Ua := {x ∈
 S
a a∈A (Ua )
λ : kxka < 1}, and k k∗a is the Minkowski-functional for (Ua )o by 1.24 .

4.21 c0 (A)∗∗ ∼
= λ∞ (A) (See [MV92, 27.14 p.314]).

Proof. By 4.20 the family (k kb )b∈λ∞ (A) is a basis of seminorms for c0 (A)∗ and

= y ∈ KN : kykb :=
n o
c0 (A)∗ ∼
X
|yj bj | < ∞ ∀b ∈ λ∞ (A) . ⇒
j∈N
∞
yj bj is in c0 (A)∗∗ .
P
(⊇) ∀b ∈ λ (A): y 7→ j∈N

(⊆) Let x ∈ c0 (A)∗∗ : ∀y ∈ c0 (A)∗ : y = j yj ej . ⇒

P
X  X X
x(y) = x yj ej = yj x(ej ) = yj xj .
j j
| {z } j
=:xj

The family is a basis of the bornology for c0 (A)∗ (see 4.18.1 ). ⇒

(Uao )a∈A
1.24 nX y X yj o
j
∀a ∈ A : ∞ > sup |x(y)| ====== sup aj xj : ≤ 1 = sup |xj aj |,
y∈Uao j
aj j
aj j

i.e. x ∈ λ∞ (A).
Thus c0 (A)∗∗ = λ∞ (A) as linear spaces and, by the closed graph theorem, also as
lcs.

66 andreas.kriegl@univie.ac.at c July 1, 2016

Duals of Köthe sequence spaces 4.23

4.22 Reflexivity of λ∞ (A) (See [MV92, 27.15 p.315]).

Let A be countable. Then
1. c0 (A) = λ∞ (A);
⇔ 2. c0 (A) is (M);
⇔ 3. c0 (A) is reflexiv;
⇔ 4. λ∞ (A) is reflexiv.

Proof. ( 1 ⇒ 2 ⇒ 3 ) by 3.28 and 3.22 .

( 3 ⇒ 4 ) by 4.21 λ∞ (A) = c0 (A)∗∗ and duals of reflexive spaces are reflexiv by
4.26 .
3.17 , see also 3.16
( 4 ⇒ 1 ) c0 (A) ⊆ λ∞ (A) closed =================
⇒ c0 (A) semi-reflexive ⇒ c0 (A) =
c0 (A)∗∗ = λ∞ (A), by 4.21 .

For 1 < p < ∞ the space λp (A) is reflexive by 3.20 and thus distinguished by
4.10 . What about λ1 (A)?

4.23 Distinguishedness of λ1 (A) (See [MV92, 27.17 p.316]).

Let A = {a(k) : k ∈ N} be countable and R+ := {t ∈ R : t > 0}.
Then λ1 (A) is distinguished ⇔

⇔ ∀D : N → R+ ∃D0 : N → R+ ∀C > 0 ∀n ∃n0 ∀j :

(k) n a(k)
n
(n) aj o j
o
min Caj , sup 0 ≤ max : k ≤ n0 .
k∈N Dk Dk

Proof. Since E := λ1 (A) is Fréchet, E ∗ has a countable basis {Uno : n ∈ N} of its

bornology.
S Hence a basis of the bornivorous disks is given by the absolutely convex
hulls of k εk Uko with ε : N → R+ . Thus E ∗ is bornological iff

∀ε : N → R+ ∃b ∈ λ∞ (A) : (Bb1 )o ⊆
D[ E
(1) εk Uko (by 2.10 ).
abs.conv.
k

(⇒) Let D : N → R+ , εk := 1/Dk ⇒ ∃b ∈ λ∞ (A) as in 1 = ⇒ ∃D0 : N → R+ :

2.9
===
w.l.o.g. b : j 7→ inf k Dk0 /aj . Let C > 0, n ∈ N and ξ : j 7→ min{Caj , 1/bj } ⇒
(k) (n)

ξ ∈ (Bb1 )o = {y ∈ λ∗ : ky · bk`∞ ≤ 1} by 4.20 ⇒ ∃n0 ∀k ≤ n0 ∃ξ k ∈ εk Uko ∃λk ∈ R:

P P k o y ∗
k≤n0 |λk | ≤ 1 and ξ = k≤n0 λk ξ by 1 . By 1.24 Ua = {y : k a k∞ = kyka ≤
1}.
n o
(n) (k) (k)
X
⇒ ∀j : min Caj , sup aj /Dk0 = ξj ≤ |λk ξjk | ≤ max0 |ξjk | ≤ max0 aj /Dk .
k k≤n k≤n
k≤n0

(⇐) Let ε : N → R+ , Dk := 2k /εk ⇒ ∃D0 : N → R+ as above. Let b : j 7→

(k) 2.9 2.10 4.20
inf k Dk0 /aj = ⇒ b ∈ λ∞ (A) =
=== ⇒ Bb1 bounded. Let ξ ∈ (Bb1 )o =
==== ⇒ |ξj | ≤
==== 1
bj =
(k) (n)
supk aj /Dk0 and ∃a ∈ A :ξ ∈ ∗
E(Ua)
o ⇒ ∃C > 0 ∃n: |ξj | ≤ min{Caj , 1/bj } ≤
(k) (k )
maxk≤n0 aj /Dk for some n0 by assumption. ⇒ ∀j ∃kj ≤ n0 : |ξj | ≤ aj j /Dkj . Let
(
ξj for k = kj
ξ k : j 7→ .
0 otherwise

andreas.kriegl@univie.ac.at c July 1, 2016 67

4.27 Duals of Köthe sequence spaces

1.24
= ⇒ 2k ξ k ∈ 2k D1k Uko = εk Uko for k ≤ n0 ⇒
====
0 0
n n
X
k
X 1 k k D[ E
ξ= ξ = k
2 ξ ∈ εk Uko
2 abs.conv.
k=1 k=1 k

Since ξ ∈ (Bb1 )o is arbitrary we are done.

4.24 None-distinguishedness of λ1 (A) (See [MV92, 27.18 p.318]).

Let A = {a(k) : k ∈ N}, where the a(k) : N2 → R+ satisfy the following conditions:
(k) (0)
1. ∀i ≥ k ∀j: ai,j = ai,j .
(m) (m+1)
2. ∀m : limj→∞ am,j /am,j = 0.
Then λ1 (A) is not distinguished.

4.23
Proof. Suppose λ1 (A) is distinguished = ⇒ ∀k : Dk := 1, C := 2, n := 0
====
∃D0 : N → R+ ∃n0 ∀i, j : min{2ai,j , sup ai,j /Dk0 } ≤ max0 ai,j
(0) (k) (k)
k k≤n

(0) (0) (k)

For i := n0 we get 2an0 ,j > an0 ,j = maxk≤n0 an0 ,j by ( 1 ) and hence
(n0 +1) (k) (k) (0) (n0 )
∀j : an0 ,j /Dn0 0 +1 ≤ sup an0 ,j /Dk0 ≤ max0 an0 ,j = an0 ,j = an0 ,j ,
k k≤n

a contradiction to ( 2 ) for m := n0 .

4.25 Example (See [MV92, 27.19 p.318]).

Let ai,j := j i for k ≤ i und ai,j := j k for k > i and A := {a(k) : k ∈ N}.
(k) (k)

Then λ1 (A) is not distinguished, so (λ1 (A))∗ is (DF) but not infra-barrelled.

Semi-reflexivity and stronger conditions on dual spaces

4.26 Proposition (See [Jar81, 11.4.5 p.228]).

E reflexive ⇒ E ∗ reflexive.

Proof. By assumption δE : E → E ∗∗ is an isomorphism. Thus also (δE )∗ :

∼
(E ∗∗ )∗ → E ∗ . We claim that idE ∗ = (δE )∗ ◦ δE ∗ : E ∗ → (E ∗ )∗∗ = (E ∗∗ )∗ −=→ E ∗ :
((δE )∗ ◦ δE ∗ )(x∗ )(x) = (δE )∗ (δE ∗ (x∗ ))(x) = δE ∗ (x∗ )(δE (x)) = δE (x)(x∗ ) = x∗ (x).
So δE ∗ = ((δE )∗ )−1 = ((δE )−1 )∗ is an isomorphism.

4.27 Proposition (See [Jar81, 11.5.4 p.230]).

E Montel ⇒ E ∗ Montel.

Proof. Let E be Montel and B ⊆ E ∗ bounded. Thus B is equicontinuous (since

E is infra-barrelled by definition) and therefore relatively compact with respect
to τpc (E ∗ , E) by the Alaŏglu-Bourbaki Theorem 3.4 . Since E is semi-Montel
τpc (E ∗ , E) = β(E ∗ , E), so E ∗ is semi-Montel.
Since E is reflexive by 3.22 , the dual E ∗ is reflexive by 4.26 and hence is
(infra-)barrelled by 4.10 . Together this shows that E ∗ is Montel.

68 andreas.kriegl@univie.ac.at c July 1, 2016

Semi-reflexivity and stronger conditions on dual spaces 4.31

4.28 Proposition. Schwartz versus quasi-normable spaces

(See [Kri07a, 6.5], [Jar81, 10.7.3 p.215]).
An lcs is Schwartz iff it is quasi-normable and every bounded set is precompact.
An lcs E is called quasi-normable (see [Jar81, 10.7.1 p.214]) iff
∀U ∃V ∀ε > 0 ∃B bounded : V ⊆ B + ε U.

Note that any normed space is quasi-normable. In fact we may take V = B := U .

Proof. In the proof of 3.31 we have shown that each bounded set in a Schwartz
space is precompact.
By definition an lcs E is Schwartz iff
∀U ∃V ∀ε > 0 ∃M finite : V ⊆ M + ε U.
Thus every Schwartz space is quasi-normable. And if every bounded set B is
precompact, then there is a finite set M ⊆ E such that B ⊆ M + ε U , and we have
the converse implication.

4.29 Counter-example.
Note that E := RX is Schwartz and even nuclear for all sets X by 3.73.1 .
However, if X is uncountable then the dual E ∗ = R(X) is not quasi-normable (hence
neither Schwartz nor nuclear).
P quasi-normable. Recall that the typical seminorms on R
Suppose E ∗ were (X)
are
given by f 7→ x cx |fx | with cx ≥ 0, see [Kri07b, 4.6.1]. Thus for the seminorm
with cx := 1 for all x there exist another seminorm given by some corresponding
cx > 0 such that for all ε > 0 there is some bounded set Bε with
n X o n X o
(1) f: cx |fx | ≤ 1 ⊆ Bε + ε · f : kf k`1 := |fx | ≤ 1 .
x x
1
For some δ > 0 the set I := {x : cx ≤ δ}
has to be (uncountably) infinite. Now
choose ε = 2δ . Then Bε is contained in a finite subsum, so there is some x ∈ I
with prx (Bε ) = {0}. Since δ · ex is an element of the left hand side of 1 , there
has to exist a b ∈ Bε and an f with kf k`1 ≤ 1 with δ · ex = b + ε · f and hence
prx (b) ≥ δ − 2δ > 0, a contradiction.

4.30 Proposition (See [Jar81, 10.7.1 p.214]).

Any lcs E is quasi-normable iff ∀U ∃V ⊆ U : (U o , β(E ∗ , E)) ,→ EV∗ o is a topological
embedding.

Proof. This inclusion is continuous (and then an embedding, since EV∗ o  Eβ∗ is
continuous) iff ∀λ > 0 ∃B bounded closed absolutely convex with U o ∩ B o ⊆ λV o .
(⇐) (B + U )o ⊆ U o ∩ B o ⊆ λV o ⇒ V ⊆ λ((B + U )o )o = λ B + U ⊆ λB + 2λU .
(⇒) V ⊆ B + λU ⇒ 2V o ⊇ 2(B + λU )o ⊇ B o ∩ (λU )o ⇒ 2λV o ⊇ U o ∩ λB o .

4.31 Proposition (See [Jar81, 12.3.1 p.254]).

Let (An )n∈N be an absorbent (bornivorous) sequence of subsets in E and U a 0-nbhd
basis consisting of absolutely convexSsets.
Then the absolutely convex hulls of k≥1 Ak ∩ Uk with Uk ∈ U (resp. the absolutely
T
convex sets k≥0 (Ak +Uk )) form a basis for the finest locally convex topology, which
coincides with the given one on each Ak .
By an absorbing (resp. bornivorous) sequence (An )n∈N in an lcs E we under-
stand a sequence of absolutely convex subsets An ⊆ E with A0 := {0}, 2An ⊆ An+1 ,

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4.33 Semi-reflexivity and stronger conditions on dual spaces

and such that each finite (resp. bounded) subsets of E is absorbed by (and hence
contained in) An for some n (See [Jar81, 12.3 p.253]).
Proof. It is easy to see, that these absolutely convex hulls form a basis for a locally
convex topology τ A , which is finer than the given one and which coincides with
S the
given one on each An : In fact τ A →SE is continuous (Uk := U and use E = k Ak )
and An → τ A is continuous (An ∩ k Ak ∩ Uk ⊇ An ∩ An ∩ Uk ).
Let now τ be another topology with that property and V be an absolutely convex
S n there is a 0-nbhd Un with An ∩ Un ⊆ V , hence the
0-nbhd for τ . Thus for each
absolutely convex hull of n An ∩ Un is contained in V , i.e. τ A ≥ τ .
Remains to show T that the two bases are equivalent:
(⊇) Let V := k≥0 (Ak + Uk ). Choose Vk with ((Vk )o )o ⊆ Uk and Uk0 ⊆ i<k Vi .
T
0
T T T
Since Am ⊆ Ak for all k ≥ m, we get Am ∩ Um ⊆ k≥m Ak ∩ k<m Vk ⊆ k Ak +
0


((Vk )o )o ⊆ V , thus V contains the absolutely convex
S
S hull of m Am ∩ Um .
(⊆) Let now U be the absolutely convex hull of m≥1 Am ∩ Um . Let kn := 2n + 1.
Then An+1 ⊆ 2−n AkT n
and there exists Vn with Vn ⊆ 2−n Ukn and 2((Vn+1 )o )o ⊆ Vn .
We claim that V := n≥0 (An +Vn+2 ) ⊆ U : Let x ∈ V , i.e. x = yn +vn with yn ∈ An
Pn
(thus y0 = 0) and vn ∈ Vn+2 . Thus x = vn + i=1 xi , where xi := yi − yi−1 . So
xi ∈ Ai + Ai−1 ⊆ Ai+1 and xi = vi−1 − vi ∈ Vi+1 − Vi+2 ⊆ Vi . Hence xi ∈ Ai+1 ∩ Vi
for all 1 ≤ i ≤ n. By the properties of A := (An )n we have x ∈ An for some n,
hence x − yn = vn ∈ 2An ∩ Vn+2 ⊆ An+2 ∩ Vn+1 , thus
n
X n
X n+1
X
x= x i + vn ∈ Ai+1 ∩ Vi + An+2 ∩ Vn+1 ⊆ 2−i · (Aki ∩ Uki ) ⊆ U.
i=1 i=1 i=1

4.32 Proposition (See [Jar81, 12.3.5 p.255]).

Let E be (quasi-)countably-barrelled and (An )n∈N an absorbent (bornivorous) se-
quence of subsets in E. Let 0 < ρn % ∞. Then an absolutely convex set U is a
0-nbhd in E iff U ∩ ρn An is a 0-nbhd in ρn An for each n.

Proof. (⇐) Let U be absolutely convex and U ∩ ρn An a 0-nbhd in ρn An for

each n.TSo let Un be absolutely convex 0-nbhds in E with Un ∩ ρn An ⊆ U . Thus
V := n U ∩ ρn An + Un is an intersection of countably many closed absolutely
convex 0-nbhds. Let B ⊆ E be finite (resp. bounded). Thus B ⊆ ρAm for some
ρ > 0 and m ∈ N. Since ρn % ∞ we may assume that B ⊆ ρm Am . Choose σ ≥ 1
with B ⊆ σUk for all k ≤ m. Then
B ⊆ σ(Um ∩ ρm Am ) ⊆ σ(U ∩ ρm Am ) ⊆ σ(U ∩ ρk Ak ) for all k ≥ m.
Thus B ⊆ σ((U ∩ρk Ak )+Uk ) for all k, and hence B ⊆ σV . Since E is (quasi-)countably-
barrelled, V is a 0-nbhd. Thus it suffices to show V ⊆ 3U : Let x ∈ V . Take m with
x ∈ ρm Am . Then V ⊆ (U ∩ ρm Am ) + Um ⊆ (U ∩ ρm Am ) + 2Um , i.e. x = y + z
with y ∈ U ∩ ρm Am and z ∈ 2Um . So x − y = z ∈ (ρm Am + U ∩ ρm Am ) ∩ 2Um ⊆
2(ρm Am ∩ Um ) ⊆ 2U and hence x ∈ 3U .

4.33 Corollary (See [Jar81, 12.3.6 p.256]).

Let E be (quasi-)countably-barrelled. Then for every absorbent (bornivorous) se-
quence (An )n∈N of subsets in E the induced locally convex topology is the given
one.

Proof. Obviously the final topology induced by the An ,→ E coincides on nAn

with the given one. So every 0-nbhd U for this locally convex topology is a 0-nbhd
for the original topology by 4.32 .

70 andreas.kriegl@univie.ac.at c July 1, 2016

Semi-reflexivity and stronger conditions on dual spaces 4.36

4.34 Lemma (See [Kri07a, 3.46], [Jar81, 12.4.7 p.260]).

Every (DF) space is quasi-normable.

Proof. Let {Bn : n ∈ N} be a basis of the bornology and U = (U o )o a 0-nbhd.

Consider the equicontiuous
S sets Ak := k U o ∩ Bko = ( k1 U ∪ Bk )o .
We claim that A := k∈N Ak is equicontinuous: ∀k ∃nk ≥ k : Bk ⊆ nk U . Thus
 \   \  \ 
Bk ∩ n1k U = Bk ∩ 1
nU ∩ Bn ⊆ Bk ∩ 1
n U ∪ Bn ⊆
n≤nk n>nk n∈N
\
⊆ Bk ∩ (An )o = Bk ∩ Ao ⊆ Ao ,
n
1
S
So the absolutely convex hull of k∈N nk U ∩ Bk ⊆ Ao , thus Ao is a 0-nbhd in the
(DF)-space E by 4.31 and 4.33 , and hence A ⊆ E ∗ is equicontinuous.
We claim that U o with the topology induced from β(E ∗ , E) continuously embeds
into EV∗ o for V := Ao : For the typical 0-nbhd k1 V o in EV∗ o we have that the
β(E ∗ , E)-0-nbhd U o ∩ k1 Bko in U o satisfies
1 o 1 1 1
Uo ∩ B = Ak ⊆ A ⊆ V o .
k k k k k
Thus E is quasi-normable by 4.30 .

4.35 Proposition (See [Jar81, 11.6.1 p.231]).

A Fréchet space is Montel iff E ∗ is Schwartz.

Proof. (⇒) We use 3.33 , so for every 0-nbhd B o ⊆ E ∗ we have to find a 0-nbhd
∗∗
C o with B oo ⊆ EC oo being compact. Since E is reflexive by 3.22 , this means that

for closed bounded B ⊆ E we have to find such a C with B in EC compact. Since

E is Montel, B is compact and hence contained in the closed absolutely convex hull
of a 0-sequence (xn ) in E by 3.6 . Since E is metrizable we find λn → ∞ with
(the closed absolutely convex hull C of) {λn xn : n ∈ N} bounded by [Kri14, 2.1.6].
Then xn → 0 in EC (since pC (xn ) ≤ λ1 n ) and thus its closed absolutely convex hull
in the Banach space EC is (pre)compact and contains B.
(⇐) Since E is Fréchet, the dual E ∗ is complete, hence semi-Montel by 3.31 .
Thus every bounded=equicontinuous subset of E ∗ is relatively compact. Hence
β(E ∗ , E) = τpc (E ∗ , E) = γ(E ∗ , E) by 3.24 . Since β(E ∗ , E) ≥ µ(E ∗ , E) always
and (Eγ∗ )∗ = Ẽ = E by [Kri07b, 5.5.7] we have β(E ∗ , E) = µ(E ∗ , E). The Fréchet
space E is reflexiv, since every continuous linear functional on (E ∗ , β(E ∗ , E)) =
(E ∗ , µ(E ∗ , E)) belongs to E by definition of µ(E ∗ , E). By 3.18 and 4.26 E ∗ is
(infra-)barrelled, hence Montel by 3.31 and thus also E ∼ = (E ∗ )∗ by 4.27 .

4.36 Lemma. Schwartzification (See [Jar81, 10.4.4 p.203]).

The topology τS of uniform convergence on E-0-sequences is the finest Schwartz
topology coarser than the given one.
A sequence x∗n ∈ E ∗ is said to be an E-0-sequence, iff there exists some equicon-
tinuous set U o with x∗n → 0 in EU∗ o , i.e. x∗n is Mackey-convergent to 0 with respect
to the bornology of equicontinuous sets (E stands for equicontinuous).
We will also write ES for the Schwartzification (E, τS ) of E. Note, that the
topology τS is denoted Tc0 and ES is denoted E0 in [Jar81, 10.4.3 p.203].
Proof.
(E ≥ τS ) since E-0-sequences are equicontinuous.

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4.38 Semi-reflexivity and stronger conditions on dual spaces

τS is Schwartz by 3.33 , since for every polar Ao of an E-0-sequence x∗n there exist
λn → ∞ such that yn∗ := λn x∗n is still a 0-sequence in EU∗ o and hence (Ao )o (the
σ = β compact closure of the absolutely convex hull of {x∗n : n ∈ N}) is compact in
o , where B := {yn : n ∈ N}.
∗ ∗
E(B o)

Now let τ 0 ≤ E be a Schwartz topology and U be a closed absolutely convex 0-

nbhd with respect to τ 0 . Then by 3.33 there exists a (τ 0 -)0-nbhd V ⊆ U with
U o ⊆ EV∗ o compact and hence contained in the closed convex hull of a 0-sequence
in EV∗ o . Since V is also a 0-nbhd in E, this sequence is an E-0-sequence and hence
U = (U o )o (since U is also E-closed) is a τS -0-nbhd.

4.37 Proposition. Universal Schwartz space

(See [Kri07a, 6.26], [Jar81, 10.5.1 p.204]).
The Schwartz spaces are exactly the subspaces of products of the Schwartzification
[
of c0 , or of its completion (c ∞ ∞ 1
0 )S = (` , µ(` , ` )).

The first statement is sometimes also called Schur’s lemma, see [Jar81, p.218].
There is however no universal (F S)-space, see [Jar81, 10.9 p.218].
Proof. In fact by 4.36 Schwartz spaces have a basis of 0-neighborhoods given by
the polars V := {x∗n : m ∈ N}o of E-0-sequences (x∗n )n∈N in E ∗ . Since pV (x) =
sup{|x∗n (x)| : n ∈ N}, the map T : x 7→ (x∗n (x))n∈N defines a continuous linear
map from E → c0 and factors over ιV : E  Q EV as T = T̃ ◦ ιV with an isometric
mapping T̃ : EV Q ,→ c0 . Since (ιV )V : E ,→ V EV is an embedding, we get an
embedding E ,→ V c0 .
It is easy to see that T̃ : (EV )S → (c0 )S is continuous: Let S ∈ L(E, F ) and yn∗ → 0
∗ ∼ ∗ −1
{yn : n ∈ N}o = {S (yn ) : n ∈ N}o and S ∗ (yn∗ ) → 0
∗ ∗ ∗


in (FV ) = FV o . Then S
∗ ∗
in ES ∗ (V o ) ⊆ E(S −1 V )o , i.e. S ∈ L(ES , FS ).
It is an embedding, since T̃ ∗ : `1 → (EV )∗ = EV∗ o is a quotient map between
Banach spaces, hence every 0-sequence in the image is the image of a 0-sequence in
the domain.
Remains to show that E embeds into the reduced projective system of the (EV )S :
Obviously E = ES → limV (EV )S is continuous.
←−
Conversely, let V := {x∗n : n ∈ N}o "
M m E
∗ ∗
with xn → 0 in (EV 0 ) for some V 0 _
and A := {x∗n : n ∈ N} ∪ {0} ⊆
(EV 0 )∗ . Since ι∗V 0 (A)o = ι−1 V 0 (Ao ), we
 & ιV 0
−1 lim E o id
lim (E )
have EV 0 ⊇ Ao = ιV 0 (ιV 0 (Ao )) = ←−V V ←−V V S
ιV 0 (ι∗V 0 (A)o ) = ιV 0 (V ), a 0-nbhd in
(EV 0 )S with ιVV (Ao ) = ιVV (ιV 0 (V )) = ιV " "  x
0 0
 
V 0
ιV (V ), thus ιV : (EV 0 )S → EV is con- E V gg
o (E V S 0 ) o o id o E V0
tinuous and hence also ιV from E ⊆
limV (EV )S into EV . ιV
0
←− V

Thus the identity from (the subspace E of) limV (EV )S → limV EV is continuous.
←− ←−
]
That (c ∞ ∞ 1
0 )S = (` , µ(` , ` )) can be found in [Jar81, 10.5.3 p.206].

4.38 Proposition. Nuclearification (See [Jar81, 21.9.1 p.508]).

The finest nuclear locally convex topology coarser than the given one is the topology
τN of uniform convergence on E-nuclear sequences.
A sequence x∗n in E ∗ is called E-nuclear (cf. 3.79 ), iff for each k ∈ N there is a
0-nbhd Uk such that (nk x∗n )n∈N is a 0-sequence (or `p for 0 < p < ∞) in EU∗ o (See
k

72 andreas.kriegl@univie.ac.at c July 1, 2016

Semi-reflexivity and stronger conditions on dual spaces 4.39

[Jar81, 21.7.1 p.500]).

We will also write EN for the nuclearification (E, τN ) of E.

Proof. (E ≥ τN ) since E-nuclear-sequences are equicontinuous (take k := 0).

(τN is nuclear)
By definition the polars of E-0-sequences (an )n∈N form a 0-nbhd basis for EN . So
let such an U := {an : n ∈ N}o be given and put V := {n2P an : n ∈ N}o ⊆ U .
The mapping S : `1 → EU∗ o defined by x = (xn )n∈N 7→ n xn an 1 / `1
since any `
is obviously well-defined continuous linear and it is onto, P D
x∗ ∈ EU∗ o is in C U o for some C > 0 hence of the form x∗ = n xn an S T
with (xn )n∈N ∈ C o`1 .  
EU∗ o / / E∗ o
ι
Similarly we have T : `1 → EV∗ o defined by x 7→ n n2 xn an .
P
V
Let D : `1 → `1 be the (by 3.58 ) nuclear diagonal mapping (xn )n∈N 7→ ( n12 xn )
and ι = (ιVU )∗ : EU∗ o  EV∗ o the natural inclusion. Thus ι ◦ S = T ◦ D is nuclear
and thereby S ∗ ◦ ι∗ is nuclear by 4.45 below and in particular absolutely summing
by 3.62 . The adjoint S ∗ of a quotient mapping S between Banach spaces is a
topological embedding (use (E/F )∗ ∼ = F o ), thus ι∗ = (ιVU )∗∗ is absolutely summing
and hence also its restriction E fV → EfU , i.e. (E, τN ) is nuclear by 3.70 .
Now let τ 0 ≤ E be some nuclear topology and let U be a closed absolutely convex
τ 0 -0-nbhd, which we may assume to be the unit-ball of a Hilbert seminorm by 3.72 .
By 3.79 there exists a fast-falling ONB (en )n∈N of EU∗ o , i.e. ∀k ∃V : {nk en : n ∈
N} ⊆ V o . Let ρ := k( n1 )n∈N k`2 and an := ∗
ρ n en . Then U o ⊆ ({an : n ∈ N}o )o : For
x∗ ∈ U o define (xn )n ∈ o`1 by xn := ρ nn i . Then
hx ,e

DX E DX E X
xn an , ek = hx∗ , en i en , ek = hx∗ , en i hen , ek i = hx∗ , ek i,
n n n

i.e. x = n xn an ∈ ({an : n ∈ N}o ) .

∗ o
P
Since the sets V from above are also 0-nbhds in E, the sequence (an )n∈N is an
E-nuclear-sequence and hence U = (U o )o ⊇ {an : n ∈ N}o is a τN -0-nbhd.

4.39 Proposition (See [Jar81, 12.5.8 p.265], [Woz13, 4.42 p.85]).

An lcs is the dual of an (FM)-space iff it is (S) and a complete (DF)-space.
It is then even ultra-bornological.

Proof. (⇒) Let F = E ∗ with E an (FM)-space. Then F is a complete (DF)-space

by 4.18.1 and is (S) by 4.35 . It is then even barrelled by 3.22 and 4.10 and
hence ultra-bornological by 4.16 .
(⇐) By 4.18.2 the dual E := F ∗ of the (DF)-space F is (F) and it is (M):
Let B ⊆ E be bounded. Since E is metrizable it is enough to show that every
countable subsetT of B is relatively compact. W.l.o.g. let B = {bn : n ∈ N} and
consider Bo = n∈N (bn )o , a countable intersection of closed 0-nbhds in F . This
set Bo is bornivorous, hence a 0-nbhd in F by the infra-countably-barrelledness of
the (DF) space F : In fact, let A ⊆ F be bounded. Then Ao is a 0-nbhd in E and
hence absorbs the bounded set B. Thus Bo absorbs A. Since F is Schwartz, there
is a E-0-sequence (yn∗ ) in F ∗ with Bo ⊇ {yn∗ : n ∈ N}o by 4.36 and hence B is
contained in the (compact) closed absolutely convex hull of {yn∗ : n ∈ N}. Thus
E = F ∗ is semi-Montel and as (F) space even Montel.
Since E = F ∗ is a Montel space, β(F ∗∗ , F ∗ ) = τc (F ∗∗ , F ∗ ). Hence
β ∗ (F, F ∗ ) = β(F ∗∗ , F ∗ )|F = τc (F ∗∗ , F ∗ )|F = τS ,

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4.41 Semi-reflexivity and stronger conditions on dual spaces

where β ∗ (F, F ∗ ) denotes the topology of uniform convergence on bounded sets in

Fβ∗ and τS is the topology on F of uniform convergence on E-0-sequences:
To see the last inequality, observe that E-0-sequences are relatively β(F ∗ , F )-compact
by 3.6 , hence τS ≤ τc (F ∗∗ , F ∗ )|F . To show the converse, note that by 3.6 for ev-
ery relatively compact set A in the Fréchet space F ∗ , there is a 0-sequence (an )n∈N
in the F ∗ (which is also an E-0-sequence by 2.2 ) such that A is contained in the
closed absolutely convex hull of the an , and therefore {an : n ∈ N}o ⊆ Ao .
Since F is complete and Schwartz, it is semi-reflexive by 3.31 and 3.22 and, by
what we have just shown, it carries the topology τS = β ∗ (F, F ∗ ), which is quasi-
barrelled (A ⊆ Fβ∗ is bounded ⇔ Ao is a bornivorous barrel). Thus F is reflexive,
hence F = F ∗∗ = E ∗ .

4.40 Proposition (See [Flo71, 5.4 p.164]).

Each locally complete (LF)-space is regular.
An (LF)-space is the reduced inductive limit of a sequence of Fréchet spaces and
similarly an (LB)-space is the reduced inductive limit of a sequence of Banach
spaces.
Proof. Let B be bounded closed and absolutely convex, thus EB is a Banach space
by local completeness. By Grothendieck’s factorization theorem 2.6 EB  E
factors over some ιn : En  E = limk Ek to a continuous linear mapping EB → En ,
−→
hence B is bounded in En .

4.41 Raikov’s completeness theorem

(See [Rai59], [Flo71, 4.1 p.162], [Sch12, 2.11 p.36]).
Let E be an lcs and (An )n∈N be an absorbent sequence of subsets of E satisfying:
1. The lcs E carries the final locally convex topology with respect to (An )n∈N .
2. Every Cauchy-net in any An is convergent in E.
Then E is complete.

Proof. Let (xj )j∈J be a Cauchy net in E and U be a 0-nbhd basis of absolutely
convex sets. We claim the following:
∃n0 ∈ N ∀U ∈ U ∀j ∈ J ∃i  j : xi ∈ U + An0 .
Otherwise, ∀n ∃Un ∃jn ∀j 0  jn : xj 0 ∈
T
/ Un + An . Put V := n (Un+1 + An ). Let
x ∈ (Um + Am−1 ) ∩ Am−1 and n ≥ m, then x = x + u with x, x0 ∈ Am−1 and
0
0
u ∈ Um , thus T u = x − x ∈ 2Am−1 ⊆ An , i.e. x = 0 + x ∈ Un+1 + An . Therefore
V ∩ Am = n<m (Un+1 + An ) ∩ Am is a 0-nbhd in Am , hence a 0-nbhd in E by
1 . Since (xj )j∈J is Cauchy, there exists j ∈ J such that xj 0 − xj 00 ∈ V for all
j 0 , j 00  j. Since (An )n∈N is absorbing there exists an n with xj 00 ∈ An−1 and hence
xj 0 ∈ xj 00 + V ⊆ An−1 + V ⊆ An−1 + (Un + An−1 ) ⊆ Un + An for all j 0  j, a
contradiction.
Now consider the net x̃ : J × U → An0 ⊆ E, which assigns to each (j, U ) an element
x̃j,U := xi − u ∈ An0 with i  j and u ∈ U . This net is Cauchy and hence converges
to some x∞ in E by 2 , since for U ∈ U there exist W ∈ U with 3W ⊆ U and
j ∈ J such that xi0 − xi00 ∈ W for all i0 , i00  j. So
x̃j 0 ,U 0 − x̃j 00 ,U 00 = (xi0 − u0 ) − (xi00 − u00 ) = (xi0 − xi00 ) − u0 + u00 ∈ 3W ⊆ U
for all (j 0 , U 0 ), (j 00 , U 00 )  (j, W ) and hence u0 ∈ U 0 ⊆ W , u00 ∈ U 00 ⊆ W , i0  j 0  j,
and i00  j 00  j.
It follows that (xj )j∈J converges to x∞ : For any U ∈ U there exist W ∈ U with

74 andreas.kriegl@univie.ac.at c July 1, 2016

Semi-reflexivity and stronger conditions on dual spaces 4.44

3W ⊆ U and j ∈ J with xj 0 − xi0 ∈ W for all i0  j 0  j and x̃j 0 ,U 0 − x∞ ∈ W for

all (j 0 , U 0 )  (j, W ), and thus xj 0 −x∞ = xj 0 −xi0 +u0 + x̃j 0 ,U 0 −x∞ ∈ 3W ⊆ U .

4.42 Proposition
(See [Sch12, 2.14 p.38], [Flo71], [Rai59], cf. [Jar81, 12.5.2 p.263]).
An (LB)-space is complete if and only if it is quasi-complete.

Proof. (⇐) Let (E, τ ) = limn (En , τn ) denote an (LB)-space, and let Bn := oEn
−→
be the closed unit ball of the Banach space (En , τn ). We will apply 4.41 for
An := 2n Bn .
Since, for each n ∈ N, we may assume Bn to be continuously injected into Bn+1 ,
the sequence (An )n∈N is an absorbing sequence.
To prove 1 , let V ⊆ E be an absolutely convex set such that V ∩ An is a
0-neighborhood of (An , τ |An ) for each n ∈ N and thus also a 0-neighborhood
of (An , τn |An ). Since An is a closed 0-neighborhood of (En , τn ), we see that V ∩ An
and hence V ∩ En ⊇ V ∩ An are also 0-neighborhoods of (En , τn ). This holds
for all n ∈ N, which means that V has to be a 0-neighborhood of the inductive
limit (E, τ ).
Remains to show condition 2 , i.e. that each τ -Cauchy net contained in some An
converges in E. But this is clear by the quasi-completeness of E since the sets An
and hence their Cauchy nets are bounded. .
Let E = limn En be a reduced inductive limit with compact connecting mappings
−→
Tn : En  En+1 , i.e. which map some absolutely convex 0-nbhd Un ⊆ En to
a relative compact subsets of Tn (Un ) ⊆ En+1  E. Let Bn be the (compact)
closure of the bounded set Tn (Un ) ⊆ E. Thus Tn factors over the normed space EB
generated by B and this space is complete by 4.3 (for τ := pB and τ 0 := E|B ), since
B is compact and hence complete. Thus we can rewrite E as reduced projective
limit of a sequence of Banach spaces with compact connecting homomorphisms.

4.43 Proposition (See [Flo71, 7.5,7.6 p.170], [Sch12, 2.8 p.33]).

Let E = limn En a reduced inductive limit of a sequence of Banach spaces with
−→
compact connecting homomorphisms En → En+1 . Then the limit is complete and
regular.

Proof. In view of 4.40 it is enough to show completeness using 4.41 : Let

An := 2n oEn where w.l.o.g. oEn ⊆ oEn+1 . Obviously (An )n∈N is an absorbing
sequence. Ad 1 : Let U ⊆ E be absolutely convex with U ∩ An a 0-nbhd for each
n and hence also in the (finer) topology induced from En on An . Since An is a
0-nbhd in En , also U ∩ An is one and hence also U ∩ En ⊇ U ∩ An . Thus U is
0-nbhd for the inductive topology of E.
Remains to show 2 : So let (xj )j∈J be a Cauchy-net in An . Since An is compact
in En+1 , the sequence xj has an accumulation point x∞ in En+1 and hence also in
E. But as Cauchy-net it has to converge to x∞ .

4.44 Proposition
(See [Jar81, 12.5.9 p.266], [Woz13, 4.30 p.75], [MV92, 25.20 p.57]).
Let F be an lcs. Then
1. F is the dual of an (FS)-space;
⇔ 2. F is a bornological (DF)-space where each bounded set is relative compact
in FA for some bounded Banach-disk A;

andreas.kriegl@univie.ac.at c July 1, 2016 75

4.46 Semi-reflexivity and stronger conditions on dual spaces

⇔ 3. F is an inductive limit of a sequence of Banach-spaces with compact con-

necting mappings;
⇔ 4. F is a complete (DF)-space, is (S), and every 0-sequence is Mackey-convergent.
A space satisfying these equivalent conditions is also called Silva-space.
Proof. ( 1 ⇒ 2 ) Let F = E ∗ with E a (FS)-space. By 4.39 F is ultra-
bornological (DF). By 3.33 ∀U ⊆ E ∃V ⊆ U : U o ⊆ FV o is compact. Since
the polars of 0-nbhds are basis of the bounded sets in F (by infra-barrelledness of
E) the condition on the bounded sets in 2 is satisfied.
( 2 ⇒ 3 ) Let {Bn : n ∈ N} be a countable basis of the bornology Sof F . By
assumption we recursively find bounded S Banach-disks An ⊇ An−1 with k<n (Ak ∪
Bk ) relative compact in FAn . Obviously n∈N FAn = F , the identity FAn−1 → FAn
is a compact operator and the identity limn FAn → F is continuous. Conversely,
−→
let B ⊆ F be bounded, so there exists an k ∈ N with B ⊆ Bk , thus B is bounded in
FAk+1 and hence also in limn FAn . Therefore the identity F → limn FAn is bounded
−→ −→
and since F is bornological it is continuous.
( 3 ⇒ 4 ) Let F = limn Fn with Fn → Fn+1 being compact between Banach spaces,
−→
hence it is ultra-bornological, complete and regular by 4.43 . In particular, F has
a countable basis of bornology formed by the multiples of the unit-balls of the Fn
and thus is (DF) and (M). Moreover, every 0-sequence is bounded, hence relatively
compact in some Fn and thus Mackey-convergent. In view of 4.28 it remains to
show quasi-normability as characterized in 4.30 : For every bounded=compact set
U o in the (FM)-space F ∗ (by 4.18.1 and 4.27 ) there exists a (Mackey-)0-sequence
x∗n → 0 such that U o is contained in its closed absolutely convex hull. Let λn → ∞
be such that {λn x∗n : n ∈ N} is bounded in F ∗ and thus contained in some V o since
F is barrelled. Then x∗n → 0 in FV∗ o and hence U o is compact in FV∗ o and thus
homeomorphic to its image in (F ∗ , β(F ∗ , F )).
( 4 ⇒ 1 ) By 4.39 F = E ∗ for some (FM)-space E. By 3.34 the (FM) space E
is (S) iff it is separable (which is automatically satisfied by 3.27 ) and σ(E ∗ , E)-
convergent sequences are β(E ∗ , E)-convergent by 3.27 , hence equicontinuously=Mackey
convergent by 4 .

4.45 Lemma (See [Jar81, 17.3.6 p.379]).

Let T : E → F be nuclear between Banach spaces. Then T ∗ : F ∗ → E ∗ is nuclear.

Proof. By assumption T = n x∗n ⊗ yn with n kx∗n k kyn k < ∞. Thus

P P
X  X
T ∗ (y ∗ )(x) = y ∗ (T (x)) = y ∗ x∗n (x) yn = x∗n (x) y ∗ (yn )
n n
X X  X 
∗
= evyn (y ) x∗n (x) = evyn (y ) x∗n (x) =
∗
evyn ⊗x∗n (y ∗ )(x),
n n n
i.e. T ∗ = ∗
P
n evyn ⊗xn with
X X
k evyn k kx∗n k = kyn k kx∗n k < ∞.
n n

4.46 Proposition (See [Jar81, 21.5.1 p.491]).

The dual E ∗ of an lcs E is nuclear iff ∀B ∃B 0 : ιB
B 0 : EB → EB 0 is nuclear
f g

An lcs E satisfying these equivalent conditions is (sometimes) called co-nuclear,

see [Jar81, 21.5 p.491].

76 andreas.kriegl@univie.ac.at c July 1, 2016

Semi-reflexivity and stronger conditions on dual spaces 4.47

Proof. A typical 0-nbhd in Eβ∗ is B o for some bounded (absolutely convex and
closed) B ⊆ E.
(⇐) By assumption there is some bounded B 0 ⊇ B such that E fB → E
g B 0 is nuclear.
Then its dual mapping (EB 0 ) → (EB ) is nuclear by 4.45 . Now note that (E ∗ )B o
∗ ∗

is isometrically embedded into (EB )∗ : The inclusion EB → E induces a morphism

E ∗ → (EB )∗ , which factors over (E ∗ )B o via an embedding, since kx∗ k(EB )∗ =
sup{|x∗ (x)| : pB (x) ≤ 1} = sup{|x∗ (x)| : x ∈ B = (B o )o } = pB o (x∗ ). So the
connecting morphism from (E ∗ )(B 0 )o → (E ∗ )B o is absolutely summing as restriction
of the (by 3.62 ) absolutely summing map (EB 0 )∗ → (EB )∗ , i.e. Eβ∗ is nuclear by
3.70 .
(⇒) Let E ∗ be nuclear, so for each closed absolutely convex bounded B there
is another one B 0 , such that (E ∗ )(B 0 )o → (E ∗ )B o is nuclear. Hence the adjoint
∗∗ ∗ ∗
EB oo = ((E )B o ) → ((E ∗ )(B 0 )o )∗ = E(B
∗∗
0 )oo is nuclear and thus the restriction to

oo
EB → EB 0 (since B = E ∩ B ) is absolutely summing and a composition of 6 such
f g
maps is nuclear, see the proof of ( 6 ⇒ 7 ) in 3.70 .

4.47 Lemma (See [Jar81, 12.5.1,12.5.2 p.263]).

For (DF)-spaces E and their Schwartzification ES := (E, τS ) we have
β(E ∗ , E) = η(E ∗ , E) = η(E ∗ , ES ) = γ(E ∗ , ES ) = τc (E ∗ , E
fS ) = β(E ∗ , E
fS ).

In particular, β(E ∗ , E) = β(E ∗ , Ẽ) provided E is (DF).

The (DF) condition can be weakend to (df) in this lemma using the same proof,
but with the sharpening mentioned in 4.18 instead of Proposition 4.18 .

Proof. Note, that obviously E → ES → (E, σ(E, E ∗ )) are continuous, hence

E ∗ = (ES )∗ . We always have:

(S) 3.24 (s.-M)

β(E ∗ , E) ≤ η(E ∗ , E) ≤ η(E ∗ , ES ) == γ(E ∗ , ES ) = = τc (E ∗ , E
==== fS ) = = β(E ∗ , E
==== fS )

The first ≤ holds, since (E, η(E ∗ , E)) := limU EU∗ o and EU∗ o  Eβ∗ is continuous.
−→
The second one holds, since id : E  → ES is continuous, so the injective limit
η(E ∗ , E) has more steps than η(E ∗ , ES ).
The first equality holds since ES is (S): In fact, EU∗ o  (E ∗ , σ(E ∗ , E)) is continuous,
so id : η(E ∗ , E) := limU EU∗ o 
→ γ(E ∗ , E) (recall 3.24 ) is continuous. Conversely,
−→
let E be Schwartz, i.e. for every 0-nbhd U there exists a 0-nbhd V with U o ⊆ EV∗ o
compact by 3.33 , and hence the induced (compact) topology from EV∗ o on U o
coincides with the restriction of σ(E ∗ , E), and the inclusion from U o with this
topology into EV∗ o is continuous. Thus γ(E ∗ , E)  → η(E ∗ , E) is continuous.
The last equality holds, since ES is a complete Schwarz space, hence semi-Montel
f
by 3.31 , thus the closed bounded subsets coincide with the compact ones.
(β(E ∗ , E) ≥ η(E ∗ , ES )) Since E is (DF), Eβ∗ is (F), by 4.18 . Let x∗n → 0 in Eβ∗ ,
then x∗n is Mackey-convergent by 2.2 , so there exists a sequence λn → ∞ with
λ2n x∗n → 0 in Eβ∗ . Since the (DF)-space E is infra-c0 -barrelled, λ2n x∗n ∈ U o for some
0-nbhd U ⊆ E. Thus λn x∗n is an E-0-sequence and hence W := {λn x∗n : n ∈ N}o
is a 0-nbhd for τS (see 4.36 ). Since x∗n → 0 in EW ∗
o and hence in lim E ∗ =:
∗ ∗ ∗
−→W W o
η(E , ES ), the inclusion β(E , E) → η(E , ES ) is (sequentially-)continuous.
The particular case follows, since by the universal property E → ES ,→ E
fS factors
∗ ∗ ∗ f
over E ,→ Ẽ. Thus β(E , E) ≤ β(E , Ẽ) ≤ β(E , ES ).

andreas.kriegl@univie.ac.at c July 1, 2016 77

4.48 Semi-reflexivity and stronger conditions on dual spaces

4.48 Proposition (See [Jar81, 21.5.3 p.491]).

For metrizable and for (DF)-spaces nuclearity and co-nuclearity are equivalent.

(F )
Proof. (nuclear ⇒ co-nuclear) Let pn be an increasing sequence of seminorms
defining the topology of E such that the connectingPmorphisms Tn : Epn+1 → Epn
are nuclear, and hence admit representations Tn = k λn,k x∗n,k ⊗ yn,k with x∗n,k ∈
o(Epn+1 )∗ , yn,k ∈ o(Epn ) and λn := k |λn,k | < ∞. Now let B ⊆ E be a closed
P
bounded
n disk, σn := sup{p o ∈ B}, let ρn := max{σn , λn σn }, and set
n+1 (b) : b
C := x ∈ E : qC (x) := n p2nn(x)
P
ρn ≤ 1 . For x ∈ B we have pn (x) ≤ pn+1 (x) ≤ σn ,
P pn (x) P σn
hence n 2n ρn ≤ n 2n σn = 1, i.e. B ⊆ C. Furthermore C is bounded since
pn (C) ≤ 2n ρn . The connecting morphism EB → EC is absolutely summable, since
for arbitrary finitely many xi ∈ EB ⊆ E we have
X X XX
pn λn,k x∗n,k (xi ) yn,k


pn (xi ) = pn (Tn (xi )) ≤
i i i k
X X X
≤ |λn,k | |x∗n,k (xi )| ≤ λn sup |x∗ (xi )|
o
x∗ ∈Un+1
k i i
X X X
∗
≤ λn sup |x (xi )| ≤ λn σn sup |x∗ (xi )| ≤ ρn sup |x∗ (xi )|.
x∗ ∈σ n Bo i x∗ ∈B o i x∗ ∈B o i

1 pn (xi )
P P P ∗ ∗ o
Thus i qC (xi ) = n,i 2n ρn ≤ sup i |x (xi )| : x ∈ B and hence the
identity EB → EC is absolutely summing by 3.62 . Since S16 ⊆ A1 ⊆ N we may
assume that it is even nuclear, and hence E is co-nuclear.
(DF )
(nuclear ⇒ co-nuclear) By 4.47 Eβ∗ = Ẽβ∗ , so we may assume that E is a com-
plete nuclear (DF). That Ẽ is (DF) can be seen as follows: By 4.47 we have
β(E ∗ , Ẽ) = β(E ∗ , E) and hence is metrizable and Ẽ has a basis of its bornology
formed by closures of bounded sets in E, since for every bounded B̃ ⊆ Ẽ we find
a bounded set B ⊆ E such that the 0-nbhds B o ⊆ B̃ o and hence B̃ ⊆ ((B̃)o )o ⊆
Ẽ
(B o )o = B . That Ẽ is quasi-c0 -barrelled is obvious (recall [Kri07b, 4.10.3]).
Let {Bn : n ∈ N} be a basis of the bornology consisting of closed absolutely convex
sets with Bn+1 ⊇ 2Bn . Put En := EBn . Since E is complete (S) hence semi-(M)
and thus semi-reflexive, EB ∼ = ((E ∗ )B o )∗ via x 7→ δ(x)|(E ∗ )Bo :
This mapping is onto, for let λ : (E ∗ )B o → K
be continuous and linear, and x ∈ E be such E∗ EO
∗∗ o
that δx := δ(x) = λ ◦ ιB ∈ E , i.e. δx (B ) =
o ι B o
ι O
B ∗ B
xx 
(ι )
{δx (x∗ ) = x∗ (x) : x∗ ∈ B o } is bounded by 
∗
(E )B o  / (EB ) ∗
EB
C := kλk and thus x ∈ C (B o )o = C B ⊆ EB .
It is also injective, for let x ∈ EB be such that δx
δ(x)|(E ∗ )Bo = 0, hence 0 = δ(x) ◦ (ιB )∗ = δx :
λ
& 
E ∗ → K, hence x = 0.
K
We claim that `1 {E} = cbs- lim `1 {En }, i.e. every bounded S ⊆ `1 {E} is contained
−→
and bounded in `1 {En } for some n (recall 3.41 ):
P∞ (n)
Suppose indirectly, that for each n we find x(n) ∈ S with πn (x(n) ) := k=0 kxk kn >
2n . So there exists a finite set Fn ⊆ N with k∈Fn kxk kn > 2n . Choose ak ∈ Bno
P (n) (n)

(n) (n)
with k∈Fn |ak (xk )| > 2n . Then
P

∀n, r ∈ N ∀k ∈ Fn+r : 2r pBno (ak

(n+r) (n+r)
) ≤ pBn+r
o (ak ) ≤ 1.

78 andreas.kriegl@univie.ac.at c July 1, 2016

Semi-reflexivity and stronger conditions on dual spaces 4.50

Thus the sequence (ak )k formed by all these finite subsequences (ak )k∈Fn for n ∈ N
(n)

converges to 0 in E ∗ and hence forms an equicontinuous set A ⊆ E ∗ by the (df)-

property. Thus Ao is a 0-nbhd in E and its Minkowski functional pAo : x 7→
sup{|a(x)| : a ∈ A} is a continuous seminorm on E. Hence πAo : (xk )k∈N 7→
is a continuous seminorm on `1 {E}. Thus πAo (S) has to be bounded,
P
k pAo (x k )
in contradiction to
(n)
X X (n) (n)
πAo (x(n) ) = sup |a(xk )| ≥ |ak (xk )| > 2n .
a∈A
k k∈Fn

The canonical map `1 [En ] → `1 [E] is continuous and `1 [E] = `1 {E} by 3.70 , so
the image of S := o `1 [En ] is bounded in `1 {E} and, by what we have just shown,
even bounded in `1 {En0 } for some n0 ≥ n, i.e. the connecting mapping En → En0
is absolutely summing, hence E is co-nuclear by 4.46 .
(F )
(nuclear ⇐ co-nuclear) By assumption and 4.18.1 E ∗ is a nuclear (DF)-space.
Hence by the second part E ∗∗ is nuclear and so is E as a subspace by 3.73.2 .
(DF )
(nuclear ⇐ co-nuclear) By assumption and 4.18.2 E ∗ is a nuclear Fréchet space.
Hence by the first part E ∗∗ is nuclear. In order to apply 3.73.2 it remains to show
that δ : E → E ∗∗ is an embedding, i.e. E is infra-barrelled: The bounded=pre-
compact (since E ∗ is (S)) sets in E ∗ are contained in the bipolar of some 0-sequence
in E ∗ by 3.6 and, since E is (df) and hence quasi-c0 -barrelled, the 0-sequences
are equicontinuous, hence the topology of E (which is that of uniform convergence
on equicontinuous sets) coincides with that induced from E ∗∗ .
This proof works also for (df) instead of (DF), however the last argument shows,
that (co-)nuclear (df) spaces are infra-barrelled and in particular (DF) spaces.

4.49 Proposition (See [Kri07a, 6.31], [Jar81, 21.5.5 p.493]).

Every strict inductive limit of a sequence of nuclear Fréchet spaces is co-nuclear.

Proof. Since strict inductive limits are regular this is immediate by 4.46 .

4.50 Theorem. Density of finite dimensional operators

(See [Kri07a, 4.44], [Jar81, 18.1.1 p.398]).
Let E be a locally convex space and B be a bornology on E. We consider on the
function spaces L(E, ) the topology of uniform convergence on all sets in B, and
hence denote them by LB . Then
1. E ∗ ⊗ F is dense in LB (E, F ) for every locally convex space F ;
⇔ 2. E ∗ ⊗ F is dense in LB (E, F ) for every Banach space F ;
⇔ 3. E ∗ ⊗ E is dense in LB (E, E);
⇔ 4. idE is a limit in LB (E, E) of a net in E ∗ ⊗ E.

Proof. ( 1 ⇒ 2 ) is trivial.
( 2 ⇒ 1 ) A typical 0-neighborhood in LB (E, F ) is given by NB,V := {T : T (B) ⊆
V } with B ∈ B and V a 0-neighborhood in F . Let ιV : F  FV be the canonical
surjection. Since FV is a normed space ιV ◦ T : E → F  FV ,→ F̃V can be
uniformly approximated on B with respect to pV : FV → K by finite dimensional
operators E → F̃V by 2 . Since FV is dense with respect to pV in F̃V we may
assume that the finite operators belong to L(E, FV ). Taking inverse images of the
vector components, we may even assume that they belong to L(E, F ).
( 1 ⇒ 3 ) and ( 3 ⇒ 4 ) are trivial.

andreas.kriegl@univie.ac.at c July 1, 2016 79

4.54 Semi-reflexivity and stronger conditions on dual spaces

( 4 ⇒ 1 ) Let Ti be a net of finite dimensional operators converging to idE , then

the net T ◦ Ti of finite dimensional operators converges to T ◦ id = T .

Let E be complete and assume that the equivalent statements of 4.50 are true
for some bornology B. And w.l.o.g. let B ∈ B be absolutely convex. Since the
identity on E can be approximated uniformly on B by finite dimensional operators,
we conclude that the inclusion EB → E can be approximated by finite dimensional
operators EB → E uniformly on the unit ball of EB . Hence it has to have relatively
compact image on the unit ball by the following lemma 4.51 , i.e. B has to be
relatively compact.

4.51 Lemma.
The set K(E, F ) of compact operators from a normed space E into a complete
space F is closed in L(E, F ).
Q f
Proof. To see this use that F = limV Ff V ⊆ V FV , hence a subset K of F
←−
is relatively compact iff ιV (K) is relatively compact in Ff V for all V . Now let
Ti ∈ K(E, F ) converge to T ∈ L(E, F ) = L(E, F ). Then the ιV ◦ Ti ∈ K(E, Ff V)
converge to ιV ◦ T in L(E, FfV ). Since F
f V is a Banach spaces it can be shown as in
[Kri07b, 6.4.8] that ιV ◦ T ∈ K(E, FV ). Hence ιV (T (oE)) is relatively compact in
f
FfV and thus T (oE) is relatively compact in F .

4.52 Definition.
A complete lcs is said to satisfy the approximation property iff the equivalent
statements in 4.50 are true for the bornology B = cp of all relatively compact
subsets of E. A non-complete space E is said to have the approximation property,
iff its completion Ẽ has it. Note that the finite dimensional operators may be taken
in L(E, E) in this situation.

4.53 Remark (See [Kri07a, 4.63], [Jar81, 18.5.8 p.414]).

For a long time it was unclear if there are spaces without the approximation prop-
erty at all. It was known that, if such a Banach space exists, then there has to be a
subspace of c0 failing this property. It was [Enf73] who found a subspace of c0 with-
= L(`2 , (`2 )∗ ) ∼
out this property. In [Sza78] it was shown that L(`2 , `2 ) ∼ ˆ π `2 )∗
= (`2 ⊗
2 2
doesn’t have the approximation property. In contrast ` ⊗π ` has the approximation
property, since by [Jar81, 18.2.9 p.403] every completed projective tensor product
of Fréchet spaces with the approximation property has it. Note however, that for
Banach spaces one can show that if E ∗ has the approximation property then so does
E, see [Jar81, 18.3.5 p.407]. Due to [H.77] is the existence of a Fréchet-Montel
space without the approximation property, see [Jar81, p416].

4.54 Lemma. “Kelley-fication” of the completion (See [Kri07a, 4.76]).

The bijection (Eγ∗ )∗γ 
→ Ẽ given by Grothendiecks completeness theorem is contin-
uous, both spaces have the same compact subsets and (Eγ∗ )∗γ carries the final locally
convex topology with respect to these subsets. If Ẽ is compactly generated, and hence
in particular if E is metrizable, then we have equality.

Proof. Recall that by Grothendiecks completeness theorem [Kri07b, 7.5.7] we

have a bijection Ẽ  → Lequi (Eγ∗ , K) into the space of linear functionals, which
are continuous on each equicontinuous set U o ⊆ E ∗ with its compact topology
σ(E ∗ , E)|U o , supplied with the topology of uniform convergence on each equicon-
tinuous set. Whereas (Eγ∗ )∗γ is the same space, but with the final locally convex

80 andreas.kriegl@univie.ac.at c July 1, 2016

Semi-reflexivity and stronger conditions on dual spaces 4.56

topology induced by the inclusions of W o with their compact topology σ(W o , E ∗ )

for all 0-nbhds W ⊆ E ∗ with respect to γ(E ∗ , E) = τc (E ∗ , Ẽ) by 3.24 .
In order to show that (Eγ∗ )∗γ → Ẽ is continuous, denote with τ̃ the topology of
Ẽ, let the polars be with respect to the duality (Ẽ, E ∗ ), and consider W o for a
0-nbhd W ⊆ Eγ∗ . Since γ(E ∗ , E) = τc (E ∗ , Ẽ) there exists a compact set K ⊆ Ẽ
with W ⊇ K o . By [Kri07b, 6.4.2] the closed absolutely convex hull (K o )o of K
is precompact and hence compact in Ẽ and hence the same is true for the closed
subset Wo ⊆ (K o )o . So on Wo the (compact) topology of τ̃ coincides with that of
σ(Wo , E ∗ ), and hence (W o , σ(W o , E ∗ ))  (Ẽ, τ̃ ) is continuous.
Conversely, let now K ⊆ Ẽ be compact. Then K o is a 0-nbhd in (E ∗ , τc (E ∗ , Ẽ)) =
Eγ∗ and thus the inclusion of the (compact) equicontinuous set ((K o )o , σ((Eγ∗ )∗ , E ∗ )) 
(Eγ∗ )∗γ is continuous. Since the inclusion (K, τ̃ )  σ(Ẽ, E ∗ ) is continuous, we get
that K is compact in (Eγ∗ )∗γ and (Eγ∗ )∗γ carries the final locally convex topology with
respect to the compact sets.

4.55 Proposition. Approximation property versus ε-product

(See [Kri07a, 4.68], [Jar81, 18.1.8 p.400]).
A complete space E has the approximation property iff F ⊗ε E is dense in the
so-called ε-product F ε E := Lequi (Fγ∗ , E) for every locally convex space F .

Note that the topology of F ⊗ε E is by definition 3.44 initial with respect to the
inclusion F ⊗ E ,→ Lequi (F ∗ , E) and has in fact values in L((F ∗ , σ(F ∗ , F )), E) ⊆
L(Fγ∗ , E).
Proof. Note that F ⊗ E is mapped into L(Fγ∗ , E), since for y ∈ F we have
δ(y) ∈ (Fγ∗ )∗ by [Kri14, 5.5.7].
(⇐) Consider the following commuting diagram:
By assumption for F := Eγ∗ the inclined arrow
on the left hand side has dense image. The ar- Eγ∗ ⊗ E / Lcp (E, E)
row on the right hand side is an embedding, since pP
(Eγ∗ )∗γ → Ẽ = E is a continuous bijection and the

equi-continuous subsets in (Eγ∗ )∗γ are exactly the
Lequi ((Eγ∗ )∗γ , E)
relatively compact subsets of Ẽ = E by 4.54 .

(⇒) Let T ∈ L(Fγ∗ , E) and let a 0-neighborhood NV o ,U in this space be given. Since
T is continuous on the compact space (V o , σ(F ∗ , F )), we have that K := T (V o )
is compact in E. By assumption E ∗ ⊗ E is dense in Lcp (E, E). Hence there
exists a finite dimensional operator S ∈ L(E, E) with (idE −S)(K) ⊆ U . Then
S ◦ T : Fγ∗ → E → E is finite dimensional and since (Fγ∗ )∗ = F̃ by [Kri14, 5.5.7] it
belongs to F̃ ⊗ E and (T − S ◦ T )(V o ) = (id −S)(K) ⊆ U . Thus T − S ◦ T ∈ NV o ,U .
Hence F̃ ⊗ε E is dense in Lequi (Fγ∗ , E) and, since F ⊗ E is dense in F̃ ⊗ε E, it is
also dense in Lequi (Fγ∗ , E).

4.56 Proposition (See [Jar81, J18.2.1 p.401]).

Let E be the reduced projective limit of spaces Ej with the approximation property.
Then E has the approximation property.

Proof. We may assume that all Ej and E is complete (since taking completions
commutes with reduced projective limits, see [Jar81, 3.4.6 p.63]). Let K ⊆ E
be compact and U ⊆ E a 0-nbhd, w.l.o.g. of the form ι−1 k (Uk ) for some k ∈ J and
0-nbhd Uk ⊆ Ek . By reducedness Fk := ιk (E) is dense in Ek hence has the approxi-
mation property. So there are ai ∈ Ek∗ and xi ∈ E such that (idFk −S)(ιk (K)) ⊆ Uk

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4.59 Semi-reflexivity and stronger conditions on dual spaces

Pn
for S := i=1 ai ⊗ ιk (xi ). Thus (idE −S̃)(K) ⊆ U for the finite dimensional oper-
Pn
ator S̃ := i=1 ι∗k (ai ) ⊗ xi .

4.57 Proposition. Consequences of nuclearity

(See [Kri07a, 6.19.2], [Jar81, 21.2.2 p.483]).
Each nuclear space has the approximation property.

Proof. Since by E is a reduced projective limit of Hilbert-spaces, it satisfies the

approximation property, by 4.56 and since Hilbert spaces have the approximation
property: PLet (ei )i∈I be an orthonormal basis. Then the net of ortho-projections
PJ : x 7→ i∈J hx, ei iei with finite J ⊆ I converges pointwise to id and is equicon-
2 1/2
P

tinuous, since kpJ (x)k`2 = i∈J |hxi , ei i| ≤ kxk`2 . So it converges for the
topology τpc = τc .

4.58 Lemma (See [Kri07a, 4.70]).

For complete spaces E and F we have F ε E ∼
= E εF.

Proof. We only have to show bijectivity, since F ε E = Lequi (Fγ∗ , E) ⊆ L(F ∗ , E)

= L(F ∗ , E ∗ ; K). To every continuous T : Fγ∗ → E
embeds into the space L(F ∗ , E ∗0 ) ∼
we associate the continuous T : Eγ∗ → (Fγ∗ )∗γ (in fact every equi-continuous set U o
∗

of E ∗ is mapped to T ∗ (U o ) = {x∗ ◦T : x∗ ∈ U o } ⊆ {y ∗ : y ∗ ∈ (T −1 (U ))o }, the polar

of a 0-neighborhood in Fγ∗ ). And by Grothendieck’s completion result (See [Kri14,
5.5.7]) we are done since by the lemma 4.54 the identity (Fγ∗ )∗γ → Lequi (Fγ∗ , K) =
F̃ is continuous.

Let us consider E ∗ ⊗
ˆ ε F now. If F is complete and satisfies the approximation
property, then Eγ ⊗ε F ∼
∗ˆ
= Lequi ((Eγ∗ )∗γ , F ) by 4.55 .

4.59 Proposition (See [Kri07a, 4.73]).

If E and F are complete, E is Montel and F (or E) satisfies the approximation
property, then
ˆ εF ∼
E⊗ = E ε F := Lequi (Eγ∗ , F ) ∼
= Lb (Eβ∗ , F ),
In more detail, for complete spaces E and F we have under the indicated assump-
tions the following identities:
F app.prop. E semi-Montel
ˆ εF =
E⊗ = E ε F = Lequi (Eγ∗ , F ) ============
=========
∗
E infra-barreled Eβ bornological
= Lequi (Eβ∗ , F ) = = Lb (Eβ∗ , F ) =
============ = L(Eβ∗ , F )
============

Proof. In the first statement the first isomorphism follows from the definition
3.44 of E ⊗ε F ⊆ Lequi (Eγ∗ , F ) ,→ L(E ∗ , F ) and the approximation property (that
it hold also if E instead of F satisfies the approximation property follows from
4.58 ). And the second one follows, since Montel spaces are barrelled by 3.22 and
3.18 and since Eγ∗ = τc (E ∗ , Ẽ) = β(E ∗ , E) by 3.24 and E being semi-Montel.

Note that the strong dual of a semi-reflexive space is barreled 4.10 . If E is in

addition metrizable, then E ∗ is bornological by 4.16 , and hence we have
Lb (Eβ∗ , F ) = L(E ∗ , F ).

82 andreas.kriegl@univie.ac.at c July 1, 2016

Semi-reflexivity and stronger conditions on dual spaces 4.62

4.60 Proposition (See [Kri07a, 4.74]).

For complete spaces Eβ∗ and F we have under the indicated assumptions the follow-
ing identities:
F app.prop. E Montel
Eβ∗ ⊗
ˆ εF = = Eβ∗ ε F := Lequi ((Eβ∗ )∗γ , F ) =
========= =======
=
E reflexive E bornological
= Lb ((Eβ∗ )∗β , F ) = = Lb (E, F ) =
======== ===========
= L(E, F ),

Proof. This follows, since the strong dual Eβ∗ of a Montel space E is Montel
by 4.27 . Note that a Montel-space E is reflexive by 3.22 , i.e. (Eβ∗ )∗β = E.
Furthermore Eβ∗ = Eβ0 is complete, provided E is bornological.

4.61 Theorem (See [Kri07a, 6.32], [Jar81, 21.5.9 p.496]).

Let E and F be Fréchet spaces with E nuclear.
Then we have the following isomorphisms:
ˆ πF ∼
1. E ⊗ = E⊗ ˆ εF ∼= L(E ∗ , F );
ˆ πF ∼
2. E ∗ ⊗ = E∗⊗ ˆ εF ∼= L(E, F );
ˆ πF ∗ ∼
3. E ∗ ⊗ = E ∗ˆ
⊗ ε F ∗ ∼
= L(E, F ∗ ) ∼ ˆ π F )∗ ;
= (E ⊗

Proof. ( 1 ) Recall that we have shown in 4.59 that for complete spaces we have
ˆ εF ∼
E⊗ = L(Eβ∗ , F ) provided E satisfies the approximation property, is Montel and
∗
Eβ is bornological. These conditions are satisfied if E is a nuclear Fréchet space by
4.57 , 3.60 , 3.31 , and 4.39 .
( 2 ) Recall that we have shown in 4.60 that for complete spaces Eβ∗ and F we
ˆ εF ∼
have Eβ∗ ⊗ = L(E, F ) provided Eβ∗ satisfies the approximation property and E is
Montel and bornological. This is all satisfied if E is a nuclear Fréchet space, since
then Eβ∗ is nuclear by 4.48 .
ˆ εF ∗ ∼
( 3 ) the same argument as in ( 2 ) applies and hence E ∗ ⊗ = L(E, F ∗ ). In
general we have L(E, F ) = L(E, F ) = L(E, F ; K) = L(E, F ; K) ∼
∗ 0 ∼
= (E ⊗ˆ π F )∗ ,
since E and F are Fréchet.

4.62 Proposition (See [Jar81, 16.4.1 p.353], [Jar81, 21.8.9 p.507]).

Let B be a bornology on E 6= {0} =
6 F.
Then LB (E, F ) is Schwartz/nuclear iff EB∗ := LB (E, K) and F are Schwartz/nuclear.

Proof. (⇒) is obvious by 3.73.2 , since F and EB∗ can be considered as (comple-
mented) subspaces.
(⇐) First one shows that a 0-neighborhood basis in LB (E, F ) is given by the sets
∗
N := N{xn },{yn∗ }o := {T : |T (xn )(ym )| ≤ 1 ∀n, m}, where xn is Mackey-convergent
to 0 in E with respect to B and yn is Mackey convergent to 0 in F ∗ with respect
∗

to the bornology of equicontinuous sets, in fact the polars of these sequences form
bases by 4.36 . Without loss of generality we may replace xn by λn xn and yn∗
by µn yn∗ with λ and µ in c0 . The functionals `j,k : LB (E, F ) → K given by
T 7→ yj∗ (T (xk )) form an equicontinuous family, since N is mapped into {λ ∈ K :
|λ| ≤ 1}. Thus λk µj `j,k are Mackey-convergent to 0 with respect to the bornology of
equicontinuous subsets. Hence its polar (which is a subset of N ) is a neighborhood
in the Schwartzification τS of LB (E, F ).
The proof for nuclearity is analogous using that by 4.38 the nuclearification is
given by the topology of uniform convergence on E-nuclear sequences x∗n ∈ E ∗ .

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4.67 Semi-reflexivity and stronger conditions on dual spaces

4.63 Corollary (See [Kri07a, 6.21 p.142]).

The ε-tensor product of Schwartz spaces is Schwartz.

Proof. This follows from 4.62 since E ⊗ε F ⊆ E ε F ⊆ L(Eγ∗ , F ) and (Eγ∗ )∗γ = Ẽ
is Schwartz.

Dual morphisms

4.64 Definition. Short exact sequences.

If T ∈ L(E, F ) is an embedding then T ∗ ∈ L(F ∗ , E ∗ ) is onto by Hahn-Banach.
If T ∈ L(E, F ) is onto (or at has at least dense image) then T ∗ ∈ L(F ∗ , E ∗ )
is injective. In order ot treat both cases simultaneoulsy we can consider short
sequences of continuous linear mappinngs
0 → E → F → G → 0.
Tn−1
A sequence · · · → En−1 − → En −Tn→ En+1 → . . . is called (algebraically)
exact iff ker Tn = img Tn−1 := Tn−1 (En−1 ) for all n. It is called topologically
exact iff Tn−1 induces an isomorphism En−1 / ker Tn−1 → ker Tn of lcs for all n.
Thus a short sequence 0 → E −S→ F −Q→ G → 0 is algebraically exact iff S is
injective, img(S) = ker(Q), and Q is onto. It is topologically exact iff in addition
S is a topological embedding and Q is a quotient mapping.
Every injective mapping (embedding) S : E → F with closed image gives rise to
the short (topologically) exact sequence 0 → E −S→ F  F/ img S → 0. And every
surjective (quotient) mapping Q : F → G gives rise to the short (topologically)
exact sequence 0 → ker Q ,→ F −Q→ G → 0.

4.65 Remark.
LetQE = limj Ej be a limit. Then E can be identified with the closed subspace
of j∈J Ej formed by all x = (xj )j∈J with F(f )(xj ) = xj 0 for all f : j → j 0 .
We get a short exact sequence 0 → E ,→ j Ej  ( j Ej )/E → 0. We can
Q Q
Q
give an explicite description of the linear space ( j Ej )/E, namely the subspace of
Q Q Q
f :j→j 0 Ej formed by the image of the mapping Q : j Ej → f :j→j 0 Ej which
0 0

given by prf :j→j 0 ◦Q := F(f ) ◦ prj − pr0j . Even for projective limits of a sequence
it however not clear, whether Q is onto or is a quotient map onto its image.

4.66 Lemma.
Every short exact sequence of (F) spaces is topologically exact.

Proof. Let T : E → F be a continuous linear mapping between Fréchet spaces.

By the open mapping theorem we get: If T is onto, then it is open hence a quotient
mapping. If T is injective with closed image, then it is an homeomorphism onto its
image, hence an embedding.

4.67 Lemma (See [MV92, 26.4 p.291]).

Let 0 → E −S→ F −Q→ G → 0 be topologically exact.
∗ ∗
Then the dual sequence 0 ← E ∗ ←S − F ∗ ←Q − G∗ ← 0 is algebraically exact.

Proof. (S embedding ⇒ S ∗ onto) by Hahn-Banach.

(Q onto ⇒ Q∗ injective) obviously.
(ker Q = img S and Q quotient mapping ⇒ ker S ∗ = img Q∗ ) For y ∗ ∈ F ∗ we have:
y ∗ ∈ ker S ∗ ⇔ y ∗ ◦ S = 0 ⇔ y ∗ |img S = 0 ⇔ y ∗ |ker Q = 0 ⇔ ∃z ∗ ∈ G∗ : y ∗ = z ∗ ◦ Q =
Q∗ (z ∗ ) ⇔ y ∗ ∈ img Q∗ .

84 andreas.kriegl@univie.ac.at c July 1, 2016

Dual morphisms 4.71

Now the question arises, whether the dual of a topological short exact sequence is
also topologically exact. Since the topology on the dual space is generated by the
polars of bounded sets and (for infra-barrelled spaces) the bornology is generated
by the polars of 0-nbhds, we need to determine how polars behave under adjoint
mappings:

4.68 Lemma (See [Jar81, 6.8.2.a p.161]).

Let T : E → F be continous linear and A ⊆ E. Then
1. (T ∗ )−1 (Ao ) = T (A)o .
2. Ao ∩ img T ∗ = T ∗ (T (A)o ).

Proof. ( 1 ) (T ∗ )−1 (Ao ) = {y ∗ : ∀a ∈ A : |y ∗ (T (a))| = |T ∗ (y ∗ )(a)| ≤ 1} = {y ∗ :

∀b ∈ T (A) : |y ∗ (b)| ≤ 1} = T (A)o .
1
( 2 ) T ∗ (T (A)o ) == T ∗ ((T ∗ )−1 (Ao )) = Ao ∩ img T ∗

4.69 Definition. Special cbs-morphisms.

Among the various structures on the dual space E ∗ of an lcs E the bornology
formed by the equicontinuous subsets is most closely related to (the topology of)
E. It will thus be essential, to consider properties of morphisms between convex
bornological spaces.
A bounded linear mapping T between separated convex bornological spaces is called
a (bornological) embedding (or cbs-embedding) iff T −1 (B) is bounded for
each bounded B. Any cbs-embedding is automatically injective, since its kernel
is a bounded linear subspace hence 0. It is called (bornological) quotient
mapping (or cbs-quotient mapping) iff each bounded B has a bounded lift B 0 ,
i.e. T (B 0 ) = B. It is enough to assume T (B 0 ) ⊇ B, since then we may replace B 0
by B 0 ∩ T −1 (B). Any cbs-quotient mapping is automatically onto, since each point
is bounded, hence the inverse image is non-empty.
Let us denote the functors b ( ) : lcs → cbs given by assigning the von Neuman
bornology and t ( ) : cbs → lcs given by assigning the topology formed by the
bornivorous absolutely convex subsets. These functors are adjoint to each other,
i.e. lcs(t E, F ) ∼
= cbs(E, b F ), see [Kri07a, 3.15]. The bornological locally convex
spaces are exactly the fixpoints under t ( ) ◦ b ( ), i.e. the image of t ( ).

4.70 Lemma.
If T : E → F is an lcs-embedding, then T : b E → b F is a cbs-embedding.

Proof. Let B ⊆ E be such that T (B) ⊆ b F is bounded. Let U be a 0-nbhd in E.

By assumption there is a 0-nbhd V in F with U = T −1 (V ). Since T (B) ⊆ λV for
some λ > 0 we have B = T −1 (T (B)) ⊆ T −1 (λV ) = λU by injectivity of T . Thus
B is bounded in b E.
The converse is not true: Let F be a bornological lcs and E a (closed) lcs-subspace
which is not bornological, e.g. 4.81 . Then its bornologification Eborn has the
same bounded sets as E, is cbs-embedded in F , but does not carry the lcs-subspace
structure.

4.71 Lemma.
If T : E → F is a cbs-quotient mapping, then T : t E → t F is an lcs-quotient
mapping.

Proof. We show that T : t E → t F is an open mapping. Let U be an absolutely

convex 0-nbhd in t E. Then T (U ) is absolutely convex and bornivorous, since any

andreas.kriegl@univie.ac.at c July 1, 2016 85

4.72 Dual morphisms

bounded B ⊆ F is image of some bounded A ⊆ E, thus A ⊆ λU for some λ > 0

and hence B = T (A) ⊆ λT (U ). Hence T (U ) is a 0-nbhd in t F .

The converse is not true, as the example 4.80 (based on 3.36 and 4.79 ) shows:
A Köthe sequence space λp (A) which is (FM), but has `p as quotient, hence the
bounded unit-ball cannot be lifted, since otherwise it would be compact.

Definition. External duality functors.

Consider duality as functor ( )∗ : lcs → cbs, which maps lcs E to the dual formed
by the continuous linear functionals together with the bornology of equicontinuous
sets, and the duality ( )0 : cbs → lcs, which maps cbs E to the dual formed by the
bounded linear functionals together with the topology of uniform convergence on
the bounded sets of E.
These two dualities form a pair of adjoint functors, since
lcs(E, F 0 ) ∼
= cbs(F, E ∗ ) = cbsop (E ∗ , F ),
see [Kri07a, 3.16].
By what we have already mentioned (see [Kri07b, 7.4.11]) the canonical mapping
E ,→ (E ∗ )0 is an lcs-embedding. And also Eβ∗ ,→ (b E)0 is an embedding by definition
of β(E ∗ , E).

4.72 Proposition (See [Kri07a, 3.18]).

1. The duality ( )0 : cbs → lcs carries cbs-quotient mappings to lcs-embeddings.

2. The duality ( )∗ : lcs → cbs carries lcs-quotient mappings to cbs-embeddings.
3. Let T : E → F be continuous and linear. Then T is an lcs-embedding iff T ∗
is a cbs-quotient mapping for the equicontinuous bornologies.
4. Furthermore, T is a dense lcs-embedding iff T ∗ is a cbs-isomorphism.

Proof. 1 Since cbs quotient mappings T : E → F are onto, we conclude that

T ∗ : F 0 → E 0 is injective. Since T ∗ (T (B)o ) = T ∗ ((T ∗ )−1 (B o )) = B o ∩ T ∗ (F 0 ), by
4.68.2 , and since the sets T (B)o form a 0-neighborhood basis of F 0 , we are done.
2 Let U be an absolutely convex 0-nbhd in E. Since T : E → F is an lcs-
quotient mapping V := T (U ) is an absolutely convex 0-nbhd in F and by 4.68.1
(T ∗ )−1 (U o ) = T (U )o = V o , thus T ∗ is a cbs-embedding.
3 (⇒) Let T : E ,→ F be an lcs-embedding and U a 0-nbhd in E. Let pU be
the Minkowski-functional of U and p̃ an extension to F , i.e. p̃ ◦ T = pU , and let
V := {y ∈ F : p̃(y) ≤ 1}. Remains to show that U o ⊆ T ∗ (V o ). So let x∗ ∈ U o , i.e.
|x∗ | ≤ p. By Hahn-Banach there exists an y ∗ ∈ F ∗ with T ∗ (y ∗ ) = y ∗ ◦ T = x∗ and
|y ∗ | ≤ p̃, hence y ∗ ∈ V o .
3 (⇐) If T ∗ : F ∗ → E ∗ is a cbs-quotient map, then (T ∗ )∗ : (F ∗ )0 → (E ∗ )0 is a
topological embedding by 1 and using the embedding E ,→ (E ∗ )0 = L((E ∗ , E), K)
of [Kri07b, 7.4.11] and the commutative diagram
 / L(E ∗ , K)
E (E ∗ )0
T L(T ∗ ,K) (T ∗ )∗
   
F  / L(F ∗ , K) (F ∗ )0
shows that T is an embedding as well.

86 andreas.kriegl@univie.ac.at c July 1, 2016

Dual morphisms 4.75

4 If T is a dense lcs-embedding, then T ∗ is injective and by 3 a cbs-quotient

mapping, hence a cbs-isomorphism. Conversely, if T ∗ is a cbs-isomorphism, then
T is an lcs-embedding by 3 and since the continuous linear functionals separate
points from closed linear subspaces, T has dense image by the injectivity of T ∗ .

4.73 Remark.
Surjectivity of linear operators D, means solvability of inhomogeneous equations
D(u) = s for arbitrary s with respect to u.
For example, by the Malgrange-Ehrenpreis Theorem (see [Kri07b, 8.3.1]) every
linear partial differential operator (PDO) D := P ( 1i ∂) with constant coefficients
C ∞ (Rn ) → C ∞ (Rn ) is onto. This can be shown, by considering the formal adjoint
operator Dt := P t ( 1i ∂) : D → D and its adjoint D̃ := (Dt )∗ on the space of
distributions D∗ (see [Kri07b, 4.9]), proving the existence of a fundamental solution
ε ∈ D∗ (i.e. D̃(ε) = δ) via Fourier transform (see [Kri07b, 8.3.1]), and obtaining
the solution of D(u) = s as u := ε ? s (see [Kri07b,
P4.7.7]). Here P is a polynomial
z 7→ |k|≤m ak z k and P t is the polynomial z 7→ |k|≤m (−1)|k| ak z k .
P

In [DGC71] it is shown that every linear partial differential operator C ω (R2 ) →

C ω (R2 ) is onto, where C ω (Rn ) denotes the space of real-analytic scalar valued
functions on Rn . In contrast, the PDO ( ∂x∂ 2 ∂ 2
) + ( ∂y ) : C ω (R3 ) → C ω (R3 ) is not
onto.

4.74 Surjectivity criterium (See [MV92, 26.1 p.289]).

Let T : E → F be continuous linear between Fréchet spaces. Then
1. T is onto;
⇔ 2. T is an lcs-quotient mapping;
⇔ 3. T ∗ : F ∗ → E ∗ is a cbs-embedding,
i.e. B equicontinuous ⇒ (T ∗ )−1 (B) equicontinuous.
⇔ 4. T ∗ : b (Fβ∗ ) → b (Eβ∗ ) is a cbs-embedding,
i.e. w.r.t. the von Neumann bornologies.

Proof. ( 1 ⇒ 2 ) by the open mapping theorem.

( 2 ⇒ 3 ) is 4.72
( 3 ⇔ 4 ) since E and F are Fréchet (hence quasi-barrelled) the β-bounded sets are
exactly the equicontinuous ones.
3
( 3 ⇒ 1 ) Let U be an absolutely convex 0-nbhd ⇒ U o equicontinuous ==
⇒ T (U )o =
F not meager
(T ∗ )−1 (U o ) (by 4.68.1 ) is equicontinuous ⇒ T (U ) = (T (U )o )o 0-nbhd = ⇒
==========
T (E) not meager (and hence T is onto
S −1 by [Kri14, 4.3.6]): Suppose T (E) ⊆
(An ) with T −1 (An ) closed, hence ∃n:
S
n An with An closed. Then E = nT
int(T −1 (An )) 6= ∅. Let x ∈ int(T −1 (An )) and U be a 0-nbhd with x+U ⊆ T −1 (An ).
Then T (x) + T (U ) ⊆ An and also T (x) + T (U ) ⊆ An , i.e. the interior of An is not
empty.

4.75 Lemma of Baernstein (See [MV92, 26.26 p.303]).

Let T : E → F continuous linear between (DF) spaces, E be (M).
Then T : E → F is an lcs-embedding iff T : b E → b F is an cbs-embedding.

Proof.
(⇒) is 4.70 .

andreas.kriegl@univie.ac.at c July 1, 2016 87

4.78 Dual morphisms

(⇐) By 4.18.2 E ∗ and F ∗ are Fréchet and T ∗ ∈ L(F ∗ , E ∗ ). By 3.22 E is

reflexive. Since T ∗∗ = T : E ∗∗ = E → F ,→ F ∗∗ it follows that T ∗ is onto by
4.74 . Let U be an absolutely convex closed 0-nbhd in E. Thus U o is bounded
and hence compact in the (M)-space E ∗ . By 3.6 this can be lifted to a compact
set K ⊆ F ∗ which has to be contained in theTclosed absolutely convex hull of a 0-
sequence (yn∗ ) in the (F)-space F ∗ . Let V := n Vn with absolutely convex 0-nbhds
Vn := {yn∗ }o . The set V is bornivorous, since yn∗ → 0, and hence a 0-nbhd, since as
(DF)-space F is quasi-countably-barrelled. Since K ⊆ h{yn∗ : n ∈ N}iabs.conv. ⊆ V o ,
we get T ∗ (V o ) ⊇ T ∗ (K) = U o and hence U = (U o )o ⊇ (T ∗ (V o ))o = T −1 (V ), i.e.
T (U ) is a 0-nbhd in the trace topology on img T .

4.76 Theorem of Eidelheit (See [MV92, 26.27 p.305]).

Let E be (F) and (x∗k )k∈N linearly independent in E ∗ . Then
∀y ∈ RN ∃x ∈ E ∀k ∈ N : x∗k (x) = yk ⇔ ∀U : dim EU∗ o ∩ {x∗k : k ∈ N}


lin.sp
< ∞.

Proof. By assumption T := (x∗k )k∈N : E → KN is continuous linear. Its adjoint

T ∗ : K(N) = (KN )∗ → E ∗ is given by T ∗ (y) = k x∗k ⊗ yk , since
P
X X 
T ∗ (y)(x) = y(T (x)) = yk x∗k (x) = x∗k ⊗ yk (x).
k k
∗
Hence T is bijective onto h{x∗k : k ∈ N}ilin.sp. (since the x∗k are linearly indepen-
dent).
By 4.74 T is onto iff (T ∗ )−1 (B) is bounded in K(N) for each bounded B ⊆ E ∗ , i.e.
for each 0-nbhd U the set T (U )o = (T ∗ )−1 (U o ) = y ∈ K(N) : k x∗k ⊗ yk ∈ U o
 P

has to be bounded and hence has to be contained in some finite subsum KN . Since
T ∗ is injective, it induces a linear isomorphism
4.68.2 [
λ · T (U )o ∼
[ [
T (U )o lin.sp. = = λ · T ∗ (T (U )o ) ====== λ · U o ∩ img T ∗
λ>0 λ>0 λ>0

= EU∗ o ∩ {x∗k : k ∈ N} lin.sp.

.

(⇒) Since T (U o

) has to be contained in some KN , we have that dim EU∗ o ∩ {x∗k :
k ∈ N} lin.sp. < ∞ for each U .
(⇐) The condition implies that the closed absolutely convex set A := (T ∗ )−1 (U o ) =
T (U )o is contained in a finite dimensional linear subspace KN and contains no R+ ·x∗
for x∗ 6= 0, since otherwise T ∗ (x∗ )|U = 0 and hence T ∗ (x∗ ) = 0, thus x∗ = 0. This
implies that A is bounded, otherwise choose an ∈ A ⊆ KN with 1 ≤ kan k → ∞
and let a∞ ∈ A be an accumulation point of ka1n k an ∈ A. Then λ a∞ ∈ A for all
λ > 0 since A 3 kaλn k an → λ a∞ for kan k ≥ λ.

4.77 Corollary. (F) spaces with KN as quotient (See [MV92, 26.28 p.305]).
Let E be (F) and not Banach. Then KN is a topological quotient of E.

Proof. Let (Un ) be a falling 0-nbhd basis of E. Since E is not Banach, we may
assume that ∃x∗k ∈ EU∗ o \ EU∗ o . Then (x∗k )k is linear independent and the mapping
k k−1

Q := (x∗k )k∈N : E → KN satisfies the assumptions of 4.76 hence is onto.

4.78 Borels theorem (See [MV92, 26.29 p.305]).

∀y ∈ KN ∃f ∈ C ∞ ([−1, 1], K) ∀k ∈ N : f (k) (0) = yk .

Proof. Let kf kk := maxj≤k kf (j) k∞ and Uk := {f : kf kk ≤ 1}. Consider x∗k :

E := C ∞ ([−1, 1], K) → K given by x∗k (f ) := f (k) (0). Obviously (x∗k )k∈N is linearly

88 andreas.kriegl@univie.ac.at c July 1, 2016

Dual morphisms 4.81

independent (on monomials). For finite sequences ξ the functional j ξj x∗j ∈ E(U
∗
P
o
k)
iff ξj = 0 for all j > k (Choose f with small derivatives of order < j but high one
of order j). Thus (x∗k )k∈N is onto by 4.76 .

4.79 λp (A) with quotient `p (See [MV92, 27.22 p.320]).

Let A = {a(k) ∈ RN ×N
: k ∈ N} with ai,j ≥ 1 and ai,k = a1,k .
(1) (k) (k)
+
p p
Then λ (A) has ` as quotient for 1 ≤ p < ∞ and c0 (A) has c0 as quotient.
P 
Proof. Q : λp (A) → `p defined by Q(x) := j xi,j /2 j
is continuous and linear,
i
j p

p
since i j xi,j /2 ≤ i k(xi,j )j k`p · k( 2j )j k`q ≤ kxkp`p · 1 ≤ kx · a(1) kp`p .
1
P P P

Claim: Q is onto (we will use 4.74 ): Q∗ : `q → λp (A)∗ , y ∈ `q , x ∈ λp (A):

X X X yi
(Q∗ y)(x) = y(Qx) = yi xi,j /2j = x ⇒ Q∗ (y) = (yi /2j )i,j .
j i,j
i j i,j
2

For k ∈ N and Uk := {x : kxkk ≤ 1} we have:

4.68.1 1.24 n X o
(Q∗ )−1 (Uko ) ====== Q(Uk )o =
===== y ∈ `q : |y(Qx)| = xi,j yi /2j ≤ kxkk .
i,j
∗ −1
Let y ∈ (Q ) (Uko ) q p
⊆ ` , ξ ∈ ` , x : (i, j) 7→ ξi δj,k . Then
X X X
|y(ξ)| = ξi yi = 2k ξi yi /2k = 2k xi,j yi /2j
i i i,j
X 1/p
(k) (k)
≤ 2k kxkk = 2k |ξi ai,k |p = 2k a1,k kξk`p
i

(k) 4.74
⇒ kyk`q ≤ 2k a1,k , i.e. (Q∗ )−1 (Uko ) is bounded = ⇒ Q is a quotient mapping.
====
For c0 (A) the proof is analogous.

4.80 Counter-example for cbs-quotient mapping (See [MV92, 27.23 p.321]).

Let A be as in 3.36 , 1 ≤ p < ∞, Q : λp (A)  `p a quotient mapping as in 4.79 .
Then Q is not a bornological quotient mapping.

Proof. The unit ball in `p is not compact and λp (A) is Montel, hence a bounded
lift would be compact.

4.81 Counter-example for inheritance of reflexivity and bornologicity

(See [MV92, 27.24 p.321]).
There is a reflexive (even (S)) ultra-bornological (DF) space with a closed not infra-
barrelled and hence not reflexive subspace.

Proof. Let λp (A) with 1 ≤ p < ∞ be the (FM) space of 3.36 . By 3.22 it is
reflexive and by 4.27 its dual E := λp (A)∗ is Montel (by 4.35 even (S)), hence
reflexive and bornological by 4.16 , and (DF) by 4.18.1 . Let Q : λp (A)  `p be
the quotient mapping as in 4.79 and consider the closed subspace F := img(Q∗ ) =
ker(Q)o in λp (A)∗ , using 4.67 . Let W be the unit ball in `p , then U := Q−1 (W )
is a 0-nbhd with Q(U ) = W . By 4.68.2 we have Q∗ (W o ) = Q∗ (Q(U )o ) = U o ∩
img Q∗ = U o ∩ F , hence Q∗ (W o ) is absolutely convex and closed in F . It is a
bornivorous barrel, since each bounded set B in F has bounded inverse (Q∗ )−1 (B)
in `q by 4.74 and hence is absorbed by the unit-ball W o . Infra-barrelledness of F
would imply that Q∗ (W o ) is a 0-nbhd in F and is bounded as image of the unit-ball.

andreas.kriegl@univie.ac.at c July 1, 2016 89

4.84 Dual morphisms

Since E is Montel it would be even relatively compact in F . Thus F would be finite

dimensional, which is a contradiction to the injectivity of Q∗ .

4.82 Surjectivity of dense mappings (See [MV92, 26.2 p.289]).

Let T : E → F be continuous linear with dense image between (F) spaces.
Then T is onto ⇔ ∀U : U o ∩ img(T ∗ ) is a Banach-disk .

Proof. By 4.68.2 Ao ∩ img(T ∗ ) = T ∗ (T (A)o ).

[Kri14, 5.4.12]+[Kri14, 5.4.17]
(⇒) U 0-nbhd ⇒ T (U ) 0-nbhd =========================⇒ T (U )o Banach-disk.
T (E) dense ⇒ T ∗ injective ⇒ ET∗ ∗ (T (U )o ) ∼
= FT∗ (U )o Banach, i.e. U o ∩ img T ∗ is a
Banach-disk.
(⇐) U 0-nbhd ⇒ B := U o ∩ img T ∗ Banach-disk, (E ∗ )B → (E ∗ , σ(ES ∗
, E)) bd. Let
(Vn )n be a 0-nbhd-basis in F ⇒ F = n Vn ⇒ (E )B ⊆ img T = n T ∗ (Vno ).
∗ o ∗ ∗
S
Vno is σ(F ∗ , F )-cp ⇒ T ∗ (Vno ) is σ(E ∗ , E)-cp ⇒ T ∗ (Vno ) ∩ (E ∗ )B is closed in (E ∗ )B .
[Kri14, 4.1.11]
= ⇒ ∃m: ∃x in the interior of T ∗ (Vmo ) ∩ (E ∗ )B ⇒ −x as well ⇒ 0 as well
===========
T ∗ inj
⇒ ∃ε > 0: εB ⊆ T ∗ (Vmo ) = ⇒ (T ∗ )−1 (B) ⊆ 1ε Vmo ⇒ (T ∗ )−1 (B) bd.
====
4.74
= ⇒ T surjective.
====

4.83 Theorem on closed image (See [MV92, 26.3 p.290], [Kri14, 9.11]).
Let T : E → F be continuous linear between (F) spaces. Then
1. img(T ) is closed;
⇔ 2. img(T ) = ker(T ∗ )o ;
⇔ 3. U o ∩ img(T ∗ ) is a Banach-disk for each 0-nbhd U ;
⇔ 4. U o ∩ img(T ∗ ) is (σ(E ∗ , E) or) β(E ∗ , E) closed for each U ;
⇔ 5. img(T ∗ ) is closed;
⇔ 6. img(T ∗ ) = ker(T )o ;
⇔ 7. T : E/ ker(T ) → img(T ) is a homeomorphism.

Proof. ( 1 ⇔ 2 ) img T = ker(T ∗ )o by [Kri14, 5.4.3].

( 1 ⇒ 7 ) T : E/ ker(T ) → img(T ) bijective continuous linear. Homeomorphism ⇔
img T closed.
( 7 ⇒ 6 ) img T ∼= E/ ker T . img(T ∗ ) ⊆ ker(T )o is obvious. Conversely, x∗ ∈
ker(T ) ⇒ ∃y ∈ (img T )∗ ∼
o ∗
= (E/ ker T )∗ , y ∗ ◦ T = x∗ ⇒ ∃z ∗ ∈ F ∗ : z ∗ |img T = y ∗ ,
i.e. T (z ) = z ◦ T = y ◦ T = x∗ .
∗ ∗ ∗ ∗

( 6 ⇒ 5 ) Obvious, since ker(T )o is closed.

( 5 ⇒ 4 ) img(T ∗ ) and U o are closed ⇒ U o ∩ img(T ∗ ) is β(E ∗ , E)-closed.
( 4 ⇒ 3 ) B := U o ∩ T ∗ (F ∗ ) closed, E ∗ = E 0 complete ⇒ (E ∗ )B complete, see
[Kri07a, 2.27].
ι [Kri14, 5.1.5]
( 3 ⇒ 1 ) Let T0 : E → T (E)(,→ F ) = ⇒ ι∗ is onto ⇒ img(T0∗ ) = img(T ∗ )
==========
4.82 , 3
= ⇒ T0 surj ⇒ T (E) = T0 (E) = T (E), i.e. img T is closed.
=======
( 4 :β(E ∗ , E)-closed⇒σ(E ∗ , E)-closed). By ( 4 ⇒ 3 ⇒ 1 ⇒ 6 ) we have U o ∩img(T ∗ ) =
U o ∩ ker(T )o , which is σ(E ∗ , E)-closed.

4.84 Exactness for dual sequence (See [MV92, 26.4 p.291]).

Let E −S→ F −Q→ G be a short sequence in (F). Then
0 → E → F → G → 0 is exact ⇔ 0 ← E ∗ ← F ∗ ← G∗ ← 0 is algebraically exact.

90 andreas.kriegl@univie.ac.at c July 1, 2016

Dual morphisms 4.87

Proof. (⇒) This is 4.67 .

(⇐) img(Q∗ ) = ker(S ∗ ) and img(S ∗ ) = E ∗ ⇒ img(Q∗ ) and img(S ∗ ) are closed ⇒
img(Q) and img(S) are closed by 4.83 .
Since img(T ∗ )o = {x : y ∗ (T x) = T ∗ (y ∗ )(x) = 0 ∀y ∗ } = ker(T ), we get

ker(S) = img(S ∗ )o = (E ∗ )o = 0
[Kri07b, 7.4.3]
img(S) = img(S) ============ ker(S ∗ )o = img(Q∗ )o = ker(Q)
[Kri07b, 7.4.3]
img(Q) = img(Q) = (img(Q)o )o ============ ker(Q∗ )o = {0}o = G.

4.85 Corollary. Duals of subspaces and quotient spaces of (F) spaces

(See [MV92, 26.5 p.292], [Kri14, 5.4.4]).
Let F ,→ E be an embedding in (F) and let 0 → (F/E)∗ → F ∗ → E ∗ → 0 be
topologically exact. Then E ∗ ∼
= F ∗ /E o and (F/E)∗ ∼
= Eo.

Proof.
By [Kri14, 5.4.4] the vertical arrows in this di-

(F/E)∗  / F∗
;
/ / E∗
OO agram are continuous bijections. The left one
 is an iso, since (F/E)∗ → F ∗ is assumed to

- "" O be an embedding and the right one is an iso,
Eo F ∗ /E o since F ∗ → E ∗ is assumed to be a quotient map-
ping.

4.86 Definition. The canonical resolution.

Let E∞ = limn En be a reduced projective limit of a sequence of Fréchet spaces
←−
with connecting morphisms fkk+1 : Ek+1 → Ek . Then the short sequence
Y Y
0 → E∞ −ι→ Ek −π→ Ek → 0,
k k
k+1
where π (xk )k∈N := xk −



fk (xk+1 ) k∈N ,

is called canonical resolution of the projective limit.

If E is a Fréchet space with basis of seminorms k kk for k ∈ N, then E is isomorphic
to the reduced projective limit formed by the Banach spaces Ek := E g Uk , where
Uk := {x ∈ E : kxkk ≤ 1} and the corresponding short exact sequence is called the
canonical resolution of the Fréchet space.

4.87 Exactness of the canonical resolutions (See [MV92, 26.15 p.299]).

The canonical resolution of any reduced projective limit of a sequence of (F) spaces
is exact. In particular, the canonical resolution of any Fréchet space is exact.

Q
Proof. Let E = limn En with (F) spaces En , i.e. E = {x ∈ k Ek : xk =
←−
fkk+1 (xk+1 ) for all k} which is the kernel of the mapping π : k Ek → k Ek
Q Q

− fkk+1 (xk+1 ))k∈N . Let ι : E ,→ k∈

Q
given by π(x)
` =∗ (x k Q N Ek be the inclusion.
∼
Obviously k Ek = ( k Ek )∗ , via (x∗k )k∈N 7→ ((xk )k∈N 7→ k x∗k (xk )). Using this
P

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4.88 Dual morphisms

identification the adjoint mappings are given by:

X X 
ι∗ (x∗ )(x) := x∗ (ι(x)) = x∗k (ιk (x)) = x∗k ◦ ιk (x) ⇒
k k
X
∗
ι : (x∗k )k∈N 7→ x∗k ◦ ιk
k

 X ∗
π ∗ (y ∗ ) (xk )k∈N := y ∗ π (xk )k∈N = yk xk − fkk+1 (xk+1 )

k
X 
yk∗ ∗ k
(xk )k∈N


= − yk−1 ◦ fk−1 ⇒
k
π ∗ : (yk∗ )k∈N 7→ (yk∗ − yk−1
∗ k
◦ fk−1 ∗
)k∈N , where y−1 := 0.
Since ι is an embedding, ι∗ is onto by Hahn-Banach. The adjoint π ∗ is injective,
since yk∗ − yk−1
∗ k
fk−1 = 0 for all k recursively gives yk∗ = 0 for all k.
= 0∗ = (π ◦ ι)∗ = ι∗ ◦ π ∗ . So let x∗ ∈ k Ek∗ with ι∗ (x∗ ) = 0. Remains
`
Obviously 0 `
to find y ∗ ∈ k Ek∗ with x∗ = π ∗ (y ∗ ), i.e.
x∗k = yk∗ − yk−1
∗ k
◦ fk−1 for all k ∈ N.
∗
Recursively we get y−1 := 0 and yk := j≤k x∗j ◦ fjk :
∗
P
 X  X
yk∗ = x∗k + yk−1
∗ k
◦ fk−1 = x∗k + x∗j ◦ fjk−1 ◦ fk−1
k
= x∗j ◦ fjk .
j≤k−1 j≤k
∗ ∗
N with x∗j
`
Since x ∈ k Ek there exists an n ∈ = 0 for all j ≥ n. For m ≥ n we
thus have
X X X 
0 = ι∗ (x∗ ) = x∗k ◦ ιk = x∗k ◦ (fkm ◦ ιm ) = x∗k ◦ fkm ◦ ιm = ym
∗
◦ ιm .
k k<n k≤m
∗
= 0 for all m ≥ n, i.e. y ∗ ∈ Ek∗ .
`
Since ιm has dense image we get ym k

Thus the dual sequence is exact and by 4.84 the canonical resolution itself is
exact.

4.88 Proposition (See [Bon91]+[MV92, 26.14 p.298]).

Let E be a Fréchet space with increasing sequence of seminorms k kk . And let k k∗k
be the Minkowski functional of the polar of Uk := {x : kxkk ≤ 1}, i.e. kx∗ k∗k =
sup{|x∗ (x)| : x ∈ Uk }. Then
1. E is quasi-normable, i.e. ∀U ∃U 0 ∀ε > 0 ∃B bd : U 0 ⊆ B + ε U .
⇔ 2. ∀p ∃p0 > p ∀q > p0 ∀ε > 0 ∃ε0 > 0 : k k∗p0 ≤ ε0 k k∗q + ε k k∗p ;
⇔ 3. ∀p ∃p0 > p ∀q > p0 ∀ε > 0 ∃ε0 > 0 : ε0 Uqo ∩ Upo ⊆ ε Upo0 .
⇔ 4. ∀p ∃p0 > p ∀q > p0 ∀ε > 0 ∃ε0 > 0 : Up0 ⊆ ε0 Uq + ε Up ;
⇔ 5. Every 0-sequence in Eβ∗ is Mackey-convergent.
∗
⇔ 6. Fborn = limn EU∗ no is sequentially retractive.
−→
We need and prove only the equivalence of the first 4 conditions.
Proof.
( 1 ⇒ 2 ) By 1 : ∀p ∃p0 > p ∀ε > 0 ∃B bd : Up0 ⊆ B + ε Up . By the boundedness
of B: ∀q > p0 ∃ε0 > 0 : B ⊆ ε0 Uq . Thus
kyk∗p0 = sup |y(x)| : x ∈ Up0 ≤ sup |y(x1 + x2 )| : x1 ∈ ε0 Uq , x2 ∈ ε Up
 

≤ sup |y(x1 )| : x1 ∈ ε0 Uq + sup |y(x2 )| : x2 ∈ ε Up = ε0 kyk∗q + ε kyk∗p .

 

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Dual morphisms 4.89

( 2 ⇒ 3 ) For given α > 0 let ε := α

2 and we have an ε0 > 0 with
kyk∗p0 ≤ ε0 kyk∗q + ε kyk∗p .
Put α0 := α
2ε0 and let y ∈ α0 Uqo ∩ Upo . Then kyk∗q ≤ α0 and kyk∗p ≤ 1, hence
kyk∗p0 ≤ ε0 · α0 + ε · 1 = α, i.e. y ∈ αUpo0 .

( 3 ⇒ 4 ) Let ε0 Uqo ∩ Upo ⊆ ε Uqo0 . Since ( ε10 Uq + Up )o ⊆ ε0 Uqo ∩ Upo we get by

polarization and bipolar theorem

o   
Uq0 ⊆ (Uqo0 )o ⊆ ε ε10 Uq +Up = ε ε10 Uq + Up ⊆ ε Uq + ε10 Uq +Up = ε00 Uq +ε Up
o
00 1
with ε := ε (1 + ε0 ).

( 4 ⇒ 1 ) [Bon91] W.l.o.g. let U := U0 and by assumption 4 (p + 1 = p0 )

∀p ∀q ∀ε > 0 ∃ε0 > 0 : Up+1 ⊆ ε0 Uq + εUp .
⇒ ∃ ε01 > 0: U1 ⊆ ε01 U2 + 2ε U0
⇒ ∃ ε̄02 > 0: U2 ⊆ ε̄02 U3 + 22εε0 U1 , i.e. ε01 U2 ⊆ ε02 U3 + 2ε2 U1 with ε02 := ε01 ε̄02 . ⇒ . . .
1
⇒ ∃ ε0k : ε0k−1 Uk ⊆ ε0k Uk+1 + 2εk Uk−1 .
Let z ∈ U1 . Then z = ε01 u2 + 2ε v1 with u2 ∈ U2 and v1 ∈ U0 and ε0k−1 uk =
P∞
ε0k uk+1 + 2εk vk with uk+1 ∈ Uk+1 T and vk ∈ Uk−1 ⊆ U0 . ⇒ ∃x := k=1 2εk vk ∈ εU0 ,
since F is Fréchet. The set B := k (ε0k + ε) Uk is bounded and z − x ∈ B, since
k ∞ ∞
 X ε  X ε 0
X ε
z−x= z− j
v j − j
v j = ε k uk+1 − vj
j=1
2 2 2j
j=k+1 j=k+1
ε
∈ ε0k Uk+1 + k Uk ⊆ (ε0k + ε)Uk .
2

4.89 Canonical resolution and quasi-normability (See [MV92, 26.16 p.299]).

Let E be (F) and π the quotient mapping of the canonical resolution of E.
If π ∗ is an embedding, then E is quasi-normable.

Proof. Indirectly assume E is not quasi-normable. (Ek )∗ ∼

= EU∗ ko and the dual
4.88
norm k kk on (Ek )∗ is just k k∗k . = ⇒
====
(1) ∃m ∀k > m ∃k 0 > k ∃εk > 0 ∀S > 0 ∃y ∈ Em ∗
: kykk > Skykk0 + εk kykm .
By assumption (π ∗ )−1 : img(π ∗ ) → k Ek∗ is continuous and p : η 7→ k εkk kηk kk
` P
` ∗
a continuous`seminorm on k Ek .
⇒ ∃p̃ SN of k Ek∗ : Pp((π ∗ )−1 (η)) ≤ p̃(η) for all η ∈ img(π ∗ ) = ker(ι∗ ).
⇒ ∃Dk ≥ 0: p̃(η) ≤ k Dk kηk kk .
1 ⇒ ∀k > max{m, Dm } ∃k 0 > k ∃εk > 0 for S := Dk0 εk /k ∃y ∈ Em ∗
with
kykk > Skykk0 + εk kykm
∗ ∗
Let η ∈ ker(ι ) be given by ηm := y, ηk0 := −y ∈ Em ⊆ Ek∗0 and ηj = 0 otherwise.
Proof of 4.87
⇒ (π ∗ )−1 (η) = ( j≤k ηj )k . Since m < k < k 0 we have:
P
===========
=
k X  
≤ p (π ∗ )−1 (η) ≤ p̃(η) ≤ Dm kykm + Dk0 kykk0


kykk ≤ p ηj
εk k
j≤k
εk εk
⇒ kykk ≤ Dm kykm + Dk0 kykk0 ≤ εk kykm + Skykk0 ,
k k
a contradiction.

andreas.kriegl@univie.ac.at c July 1, 2016 93

4.91 Dual morphisms

4.90 Sequences of bounded subsets in (F) spaces (See [MV92, 26.6 p.292]).
Let E be metrizable. Then
S
1. Bn ⊆ E bounded ⇒ ∃δn > 0: n∈N δn Bn bounded.
2. ∀ B bounded ∃ C ⊇ B bounded: EC ⊇ B → E is an embedding.

Proof. Let (Un ) be a 0-nbhd basis of E.

( 1 ) ∀j, n ∃λj,n > 0: Bj ⊆ λj,n Un . Let λn := max{λj,n : j ≤ n} ⇒ ∀j ≤ n :
λj,n ≤ λnT⇒ ∀j ∃αj ≥ 1 ∀n:Tλj,n ≤ αj λn ⇒ ∀j, n: Bj ⊆Sλj,n Un ⊆ αj λn Un ⇒ ∀j:
Bj ⊆ αj n λn Un and B := n λn Un is bounded. Thus j α1j Bj ⊆ B is bounded.
T
( 2 ) W.l.o.g. B absolute convex. ∃λn > 0 : B ⊆ λn Un ⇒ B ⊆ C := n nλn Un
and C is bounded and EC ⊇ B → E continuous. In order to show the converse, let
2
T
x ∈ B and ε > 0. ∃m: ε > m , ∃k: Uk ⊆ n<m εnλn Un ⇒
\  \ \
2B ∩ Uk ⊆ εnλn Un ∩ εnλn Un = ε nλn Un = ε C
n≥m n<m n

⇒ B ∩ (x + Uk ) = x + (B − x) ∩ Uk ⊆ x + 2B ∩ Uk ⊆ x + ε C
⇒ B ∩ (x + Uk ) ⊆ B ∩ (x + ε C).

4.91 Theorem. Dual of surjective mappings (See [MV92, 26.7 p.293]).

Let Q : F  G be continuous linear between (F) and onto. Then
Q is a cbs-quotient mapping ⇔ Q∗ is an lcs-embedding.

Proof. (⇒) This is 4.72.1

(⇐) B ⊆ G bd ⇒ B o 0-nbhd ⇒ Q∗ (B o ) 0-nbhd in img(Q∗ ) ⊆ F ∗ ⇒ ∃M ⊆ F bd.
Q∗ inj
M o ∩ img(Q∗ ) ⊆ Q∗ (B o ) =====
⇒
4.68.1
Q(M )o ====== (Q∗ )−1 (M o ) = (Q∗ )−1 (M o ∩ img(Q∗ )) ⊆ B o
⇒B ⊆ (B o )o ⊆ (Q(M )o )o = Q(M )
4.90.2 for Q(M ) GB 0
⇒∃B 0 ⊆ G bd. : Q(M ) = Q(M )
==============
recursion
⇒ ∀B ⊆ G bd. ∃M ⊆ F bd. ∃B 0 ⊆ G bd. ∀ε > 0 : B ⊆ Q(M ) + εB 0 . =======
⇒
∀B0 ⊆ G bd. ∀n ∃Mn ⊆ F bd. ∃Bn ⊆ G bd. ∀ε > 0: Bn ⊆ Q(Mn ) + εBn+1 .
4.90.1
======
= ⇒ ∃M, B bd. ∃λn > 0: Mn ⊆ λn M , Bn ⊆ λn B. W.l.o.g. M closed and
λ0 = 21 . ⇒ Take ε0 := 1 and εn ≤ 1 such that εn λn ≤ 1/2n+1 .
∀b0 ∈ B0 ∀n ∃mn ∈ Mn ∃bn+1 ∈ Bn+1 : bn = Q(mn ) + εn+1 bn+1 . ⇒
X 
b0 = Q ε0 · · · εj mj + ε0 · · · εk+1 bk+1
j≤k
1
ε0 · · · εj mj ∈ e0 · · · εj Mj ⊆ εj λj M ⊆ M
2j+1
1
ε0 · · · εk+1 bk+1 ∈ e0 · · · εk+1 Bk+1 ⊆ εk+1 λk+1 B ⊆ B
2k+2
∞
X ∞
X
⇒ ∃m := ε0 · · · εj mj ∈ M and Q(m) = ε0 · · · εj (bj − εj+1 bj+1 ) = b0 ,
j=0 j=0

i.e. B0 ⊆ Q(M ).

94 andreas.kriegl@univie.ac.at c July 1, 2016

Dual morphisms 4.95

4.92 Proposition. (See [MV92, 26.11 p.295]).

Let 0 → E −S→ F −Q→ G → 0 be exact between (F).
Q a cbs-quotient mapping ⇒ S ∗ is an lcs-quotient mapping.

Proof. Let (Un )n and (Vn )n 0-nbhd-bases of F and G.

Claim. ∀n ∃m ∀B ⊆ Vm bd ∃M ⊆ Un bd: Q(M ) = B:
Indirect: ∃n
S ∀m ∃Bm ⊆ Vm bd ∀M ⊆ F bd Q(M ) = B ⇒ S M 6⊆ Un .
⇒ B := 2m Bm ⊆ G bd (in fact: 2Vn+1 ⊆ Vn , ⇒ m≥n 2m Bm ⊆ 2n Vn ) ⇒
∃M ⊆ F bd: Q(M ) = B ⇒ ∃ε > 0: εM ⊆ Un , ∃m: ε2m ≥ 1 ⇒ Q(εM ) = εB ⊇
ε2m Bm ⊇ Bm . ⇒ M0 := εM ∩ Q−1 (Bm ) ⊆ Un , Q(M0 ) = Bm , a contradiction.
Claim. ∀B ⊆ G bd ∃M ⊆ F bd ∀n ∃k: Q(M ∩ Un ) ⊇ B ∩ Vk :
∀n ∃mn ∀B ⊆S G bd ∃Mn ⊆ Un : Q(Mn ) = Bn := B ∩SVmn . S
Put M := h n Mn iabs.conv . ⇒ M ⊆ F bd, since k≥n Mk ⊆ k≥n Uk = Un .
Q(M ∩ Un ) ⊇ Q(Mn ) = Bn = B ∩ Vmn
Claim. ∀L ⊆ F bd ∃D ⊆ E bd ∀n ∃m: (L + Um ) ∩ E ⊆ D + Un
(See [MV92, 26.10 p.295]):
Let L ⊆ F bd, B := Q(L) ⇒ ∃M ⊆ F bd ∀n ∃k: Q(M ∩ Un ) ⊇ B ∩ Vk .
Put D := (L + M ) ∩ E ⊆ E bd. ∀n ∃n̄ ≥ n: 2Un̄ ⊆ Un ∃k: Q(M ∩ Un̄ ) ⊇ B ∩ Vk .
Q continuous ⇒ ∃m ≥ n̄: Q(Um ) ⊆ Vk .
Let x ∈ (L + Um ) ∩ E. ⇒ ∃l ∈ L ∃u ∈ Um : x = l − u, Q(l) − Q(u) = Q(x) = 0.
Q(l) = Q(u) ∈ B ∩ Q(Um ) ⊆ B ∩ Vk ⊆ Q(M ∩ Un̄ ) ⇒
∃ξ ∈ M ∩ Un̄ : Q(ξ) = Q(l) = Q(u) ⇒
x = (l − ξ) + (ξ − u) ∈ (L + M ) ∩ E + Un̄ + Um ⊆ (L + M ) ∩ E + 2Un̄ ⊆ D + Un

Claim. S ∗ is a quotient mapping (See [MV92, 26.9 p.295]):

∀L ⊆ F bd ∃D ⊆ E bd ∀n ∃m: (L + Um ) ∩ E ⊆ D + Un .
Let y ∈ Do . ⇒ ∃n: y ∈ Uno ⇒ y ∈ Do ∩ Uno ⊆ 2(D + Un )o ⇒ y ∈ 2((L + Um ) ∩ E)o
Hahn Banach
= ⇒ ∃ỹ ∈ 2(L + Um )o ⊆ 2Lo : ỹ ◦ S = y, i.e. S ∗ (Lo ) ⊇ 21 Do .
==========

4.93 Theorem. Exactness of dualized sequence (See [MV92, 26.12 p.296]).

Let 0 → E → F −Q→ G → 0 be exact between (F) spaces. Then
0 ← E ∗ ← F ∗ ← G∗ ← 0 is topologically exact ⇔ Q is bornological quotient
mapping.

Proof. 4.84 , 4.91 , 4.92 ⇒

4.94 Lifting compact sets along quotient mappings in (F)

(See [MV92, 26.21 p.302]).
Let Q : E → F be continuous linear surjective between (F) spaces.
Then Q is a cbs-quotient mapping for the bornologies of relatively compact subsets.

⇒ ∃xn → 0, K ⊆ h{xn : n ∈ N}i ⇒ ∃zn → 0 in E with

3.6 (!)
Proof. K ⊆ F compact =
===

⇒ L := h{zn : n ∈ N}i cp, Q(L) ⊆ K.

3.6
===
Q(zn ) = xn =

4.95 Theorem. Dual of sequences in (F) with (M) quotient (See [MV92,
26.22 p.303]).
Let 0 → E → F → G be a sequence in (F) and let G be (M). Then
0 → E → F → G → 0 is exact ⇔ 0 ← E ∗ ← F ∗ ← G∗ ← 0 is topologically exact.

andreas.kriegl@univie.ac.at c July 1, 2016 95

4.97 Dual morphisms

Proof. G (M) ⇒ bounded sets are relatively compact, hence have bounded lifts
along Q by 4.94 . Thus Q is a cbs-quotient mapping, hence the dual sequence is
topologically exact by 4.93 . The converse follows from 4.84 .

4.96 Exactness for quasi-normable spaces (See [MV92, 26.13 p.296]).

Let 0 → E → F −Q→ G → 0 be exact in (F) and let E quasi-normable.
Then Q is a cbs-quotient mapping.

4.88.4
⇒ ∃nk , Uk := Wnk , Vk := Uk ∩ E:
Proof. (Wn )n 0-nbhd-basis of F . ======
(1) ∀k ∀ε > 0 ∃ε0 > 0 : Vk ⊆ ε0 Vk+1 + ε Vk−1 .
Q open
Let B ⊆ G bd. = =====⇒ ∀k ∃Ck : B ⊆ Ck Q(Uk ). Put Ck0 := Ck + Ck+1 . We use
recursion to construct
1
(2) ∀k ≥ 2 ∃εk > 0 : Vk ⊆ εk Vk+1 + k Vk−1 with Dk := Ck0 + Dk−1 εk−1 .
2 Dk
In fact, put D1 := 0, and in the induction step let ε := 1/(2k Dk ) and take εk := ε0
as in 1 .
For k kk := pUk let M := {x ∈ F : ∀k ≥ 2 : kxkk ≤ Ck + Dk εk + Ck0 + 1}. Then M
is bounded.
Claim: Q(M ) ⊇ B.
Let ξ ∈ B ⊆ Ck Q(Uk ) = Q(Ck Uk ) ⇒ ∃xk ∈ Ck Uk , Q(xk ) = ξ. Put yk :=
xk − xk+1 . ⇒ Q(yk ) = Q(xk − xk+1 ) = ξ − ξ = 0, i.e. yk ∈ E = ker(Q) and
yk = xk − xk+1 ∈ Ck Uk + Ck+1 Uk+1 ⊆ (Ck + Ck+1 ) Uk = Ck0 Uk
⇒ ∀k : yk ∈ Ck0 Uk ∩ E = Ck0 Vk .
We use induction to construct vk ∈ Dk εk Vk+1 and uk ∈ 2−k Vk−1 such that
yk + vk−1 = vk + uk .
Let v0 := 0 and vk−1 already be given. Then yk + vk−1 ∈ (Ck0 + εk−1 Dk−1 )Vk =
2
⇒ ∃vk ∈ Dk εk Vk+1 , uk ∈ 2−k Vk−1 : yk + vk−1 = vk + uk .
Dk Vk ==
1
P P P P
⇒ ∃bk := vk−1 − j≥k uj ∈ E, since j>k kuj kk ≤ j>k kuj kj−1 ≤ j>k 2j =
1
2k
.
X 1
kbk kk = vk−1 − uk − uj k ≤ kvk kk + kyk kk + k ≤ Dk εk + Ck0 + 1.
| {z } 2
j>k
=vk −yk

Let x := x2 + b2 . Since bk+1 − bk = vk − vk−1 + uk = yk = xk − xk+1 , we have

kxkk = kxk + bk kk ≤ kxk kk + kbk kk ≤ Ck + (Dk εk + Ck0 + 1) ⇒ x ∈ M and
Q(x) = Q(x2 ) + Q(b2 ) = ξ + 0.

4.97 Theorem. Dual sequences and quasi-normability (See [MV92, 26.17

p.300]).
Let E be (F). Then
1. E is q-normable
⇔ 2. If Q : F  G is an lcs-quotient mapping in (F) with kernel E, then Q is a
cbs-quotient mapping.
⇔ 3. If 0 → E → F → G → 0 is exact in (F), then 0 ← E ∗ ← F ∗ ← G∗ ← 0 is
topologically exact.

96 andreas.kriegl@univie.ac.at c July 1, 2016

Dual morphisms 4.102

Proof. ( 1 ⇒ 2 ) by 4.96 .
( 2 ⇒ 3 ) by 4.93 .
( 3 ⇒ 1 ) 4.89 for the canonical resolution of 4.87 .

4.98 Corollary. Quasi-normable (F) are distinguished

(See [MV92, 26.18 p.301]).
q-normable (F) ⇒ distinguished.

4.97
Proof. = ====⇒ dual of canonical resolution of 4.87 is topological exact. ⇒ E ∗ is
quotient of countable coproduct of Banach spaces, hence bornological by 2.5 .

4.99 Dual sequences for Schwartz spaces (See [MV92, 26.24 p.303]).
Let 0 → E → F → G → 0 be exact in (F) and let one of these 3 spaces be in (S).
Then the dual sequence is topologically exact.

Proof. Closed subspaces and quotients of (FS) are (FS) by 3.73 , hence (M) and
quasi-normable. Thus 4.97 and 4.95 yield the result.

4.100 Corollary (See [MV92, 26.25 p.303]).

Let F be (FS) and E ⊆ F be closed. Then E ∗ ∼
= F ∗ /E o and (F/E)∗ ∼
= Eo.

Proof. 4.99 , 4.85 ⇒

4.101 Example (See [MV92, 27.19 p.318]).

Let A = {a(k) : k ∈ N} as in 4.25 . Then the Fréchet space λ1 (A) is not distin-
guished, not quasi-normable and (λ1 (A))∗ is (DF) but not infra-barrelled.

Proof. In 4.25 we have shown that λ1 (A) is not distinguished and hence (λ1 (A))∗
is (DF) but not infra-barrelled. By 4.98 λ1 (A) is not quasi-normable.

4.102 Quasi-normability of λp (A) (See [MV92, 27.20 p.318]).

For A = {a(n) : n ∈ N} let E := λp (A) with 1 ≤ p < ∞ oder E := c0 (A). Then
1. E is q-normable
(p0 ) (q) (p)
⇔ 2. ∀p ∃p0 ∀ε > 0 ∀q ∃ε0 > 0 ∀j : 1/aj ≤ ε0 /aj + ε/aj .
⇔ 3. ∀p ∃p0 ∀J ⊆ N: inf j∈J aj /aj
0
(p) (p ) (p) (q)
> 0 ⇒ ∀q ≥ p0 : inf j∈J : aj /aj > 0.

Proof. E = λp (A) (For E = c0 (A) analogously)

4.88
( 1 ⇒ 2 ) Up := {x ∈ λp (A) : kxkp ≤ 1}. = ⇒
====
∀p ∃p0 > p ∀ε > 0 ∀q ∈ N ∃ε0 > 0 ∀y ∈ E ∗ : kykUpo0 ≤ ε0 kykUqo + εkykUpo

(p) (p) 1.24

( 2 ) is obvious for j with aj = 0. Otherwise aj 6= 0 = ⇒ ej ∈ hUpo i and
====
(p0 ) (q) (p)
1/aj = kej kUpo0 ≤ ε0 kej kUqo + εkej kUpo = ε0 /aj + ε/aj .

( 2 ⇒ 3 ) ∀p ∃p0 satisfying ( 2 ). I ⊆ N (p)

with inf j∈I aj /aj
(p0 )
=: η > 0. Put
0 0 (p) (p0 ) (p) (q)
ε := η/2. For q ≥ p ∃ε > 0 satisfying aj /aj ≤ ε0 aj /aj + ε for all j ⇒
(p) (q) (p) (p0 ) 0 η
aj /aj ≥ (aj /aj − ε)/ε ≥ 2ε0 .

andreas.kriegl@univie.ac.at c July 1, 2016 97

4.102 Dual morphisms

( 3 ⇒ 1 ) ∀p ∃p0 > p satisfying ( 3 ). Let ε > 0, q ∈ N. Put I := {j : aj /aj

(p) (p0 )
≥ ε}
( 3 ) (p) (q)
= ⇒ ∀q ≥ p0 ∃ε0 > 0: inf j∈I aj /aj
=== ≥ 1
ε0 .
4.88.4
Claim: Up0 ⊆ ε0 Uq + εUp (= ⇒ λp (A) q-normable):
======
Let x ∈ Up0 and z := x − y with yj := xj for j ∈ I and 0 otherwise. ∀j ∈ I:
(p0 ) (p) (q)
aj ≥ aj ≥ aj /ε0 ⇒
0 1
kykq = y · a(p ) · · a(q) ≤ kxkp0 ε0 ≤ ε0
a(p0 ) `p
(p) (p0 )
∀j ∈
/ I: aj < εaj ⇒
0 1
kzkp = z · a(p ) · · a(p) ≤ kxkp0 ε ≤ ε
a(p0 ) `p
⇒ x = y + z ∈ ε0 Uq + εUp .

98 andreas.kriegl@univie.ac.at c July 1, 2016

Dual morphisms 4.107

Splitting sequences

In this section we describe situations, where short exact sequences split and refor-
mulate this in terms of the derived functor Ext of the Hom-functor. For sequences
with a power series space as kernel the characterizing property on the quotient is
(DN). And for sequences with a power series space of infinite (resp. finite) type
as quotient the characterizing property on the kernel is (Ω) (resp. (Ω)). We show
that the spaces in (DN) are exactly the subspaces of generalized power series spaces
λ∞
∞ (a) of infinite type. And the spaces in (Ω) are exactly the quotients of gener-
alized power series spaces λ1∞ (a) of infinity type. We give some applications to
extensions of non-linear mappings and introduce universal linearizer for that.

4.103 Continuously solving PDE’s.

In the early fifties of the 20th century L. Schwartz posed the problem of determining
when a linear partial differential operator P (∂) has a (continuous linear) right
inverse. Grothendieck has shown that for n ≥ 2 no elliptic operator has such an
inverse on C ∞ (U ) for U ⊆ Rn . By [Vog84, Theorem 3.3 p.365] the same holds
more generally for hypoelliptic operators. This is in contrast to that fact, that
hyperbolic PDO’s have continuous linear right inverses on C ∞ (Rn ).
A partial differential operator D defined on an open subset U ⊆ Rn is called
hypoelliptic if for every distribution u defined on an open subset V ⊆ U such
that Du is C ∞ is itself C ∞ . The Laplacian ∆ is elliptic and thus hypoelliptic. The
∂
heat equation operator D(u) := dt u − ∆(u) is hypoelliptic (even parabolic) but not
∂ 2
elliptic. The wave equation operator D(u) := ( dt ) u − ∆(u) is not hypoelliptic,
it is hyperbolic; i.e. the Cauchy problem is uniquely solvable in a neighborhood of
each point p for any initial data given on a non-characteristic hypersurface passing
through p.

4.106 Continuously extending functions or jets.

The exact sequence
f ∈ C ∞ ([−1, 1]) : f is ∞-flat at 0 ,→ C ∞ ([−1, 1])  RN


of Borel’s theorem 4.78 is not splitting. Otherwise, there would be an embedding

RN ,→ C ∞ ([−1, 1]) and the ∞-norm on C ∞ ([−1, 1]) would induce a seminorm on
RN with kernel {0}. But each seminorm x 7→ maxi≤n |xi | of the usual basis of
seminorms for the product RN has an infinite dimensional kernel {x ∈ RN : xi =
0 for all i ≤ n}.
For subsets ι : A ⊆ Rn let us consider the property, that the restriction operator
ι∗ : C ∞ (A, R)  C ∞ (Rn , R) has a continuous linear right inverse.
By [See64] Rt≥0 ⊆ R has this property and by [Tid79, Satz 4.5 p.308] for each
1 } ⊆ R has it.
r > 1 the set A := {x ∈ Rn : 0 ≤ x1 ≤ 1, x22 + · · · + x2n ≤ x2r n

Whereas, by [Tid79, Beispiel 2 p.301] the set A := {(x, y) : x ≥ 0, |y| ≤ ϕ(x)} does
not have it, when ϕ ∈ C ∞ (R, R) is ∞-flat at 0.

4.107 Proposition.
The functor L( , F ) : lcsop → lcs is left exact,
i.e. if 0 ← E + ←Q− E ←S− E − is topologically exact, then
∗ ∗
0 → L(E + , F ) −Q → L(E, F ) −S → L(E − , F )
is also exact, i.e. L( , F ) is a left exact functor.

Proof. (Q∗ is injective) Let 0 = Q∗ (ϕ) = ϕ ◦ Q. Then ϕ = 0, since Q is onto.

andreas.kriegl@univie.ac.at c July 1, 2016 99

4.110 Splitting sequences

(ker(S ∗ ) = img(Q∗ )) Let 0 = S ∗ (ϕ) = ϕ ◦ S, i.e. ϕ vanishes on img(S) = ker(Q)

and hence factors to a ϕ̃ : E + → F with ϕ = ϕ̃◦Q = Q∗ (ϕ̃). The converse inclusion
is obvious by Q ◦ S = 0.

The functor L( , F ) is not exact (i.e. maps short exact sequences to such sequences)
in general, since exactness at L(E − , F ) would mean that for closed embeddings
S : E − ,→ E the adjoint S ∗ : L(E, F ) → L(E − , F ) is onto, i.e. every morphism
ϕ : E − → F must have an extension to E.

4.108 Definition. Injective spaces and extension of maps.

An (F) space E is called injective Fréchet space iff for every
 S /
H
embedding S : H ,→ G of (F) spaces every T ∈ L(H, E) has an
G
extension T̃ ∈ H(G, E), i.e. T̃ ◦ S = T .
Thus the Fréchet spaces F for which L( , F ) preserves exactness T̃
of all short exact sequences in (F) are exactly the injective ones.
T 
E
By Hahn-Banach K is an injective (Fréchet) space.
More generally, for every set X the Banach space `∞ (X) of bounded functions on X
is an injective Fréchet space: Let S : H ,→ G be an embedding and T : H → `∞ (X)
be continuous, i.e. p := k k`∞ ◦ T is a continuous seminorm on H and hence has
an extension to a continuous seminorm p̃ on G. Now Tx := evx ◦T ∈ H ∗ for each
x ∈ X and |Tx (y)| ≤ kT (y)k∞ = p(y) for all y ∈ H. By Hahn Banach we find
T̃x ∈ G∗ with |T̃x (z)| ≤ p̃(z) for all z ∈ G. Thus T̃ : G → `∞ (X) defined by
T̃ (z)(x) := T̃x (z) for each z ∈ G and x ∈ X is a continuous linear extension of T ,
since kT̃ (z)k∞ = sup{|T̃x (z)| : x ∈ X} ≤ p̃(z) < ∞ for all z ∈ G.
Every Fréchet space F is subspace of an injective Fréchet space: We can embed F
into a countable product of Banach spaces and every Banach space G can be embed-
ded into the space of bounded linear functionals on G∗ and thus into `∞ (oG∗ ). Since
a countable product of injective Fréchet spaces is obviously an injective Fréchet
space we are done.

4.109 Proposition.
The functor L(E, ) : lcs → lcs is also left exact.

Proof. Let 0 → F − −S→ F −Q→ F + be topologically exact and consider

0 → L(E, F − ) −S∗→ L(E, F ) −Q∗→ L(E, F + ).

It is exact at L(E, F − ), since S∗ is obviously injective.
It is exact at L(E, F ), since ϕ ∈ L(E, F ) is in ker(Q∗ ) ⇔ 0 = Q∗ (ϕ) = Q ◦ ϕ
⇔ img(ϕ) ⊆ ker(Q) = img(S) ⇔ ϕ factors to a morphism ϕ̃ : E → F − over the
embedding S : F − ,→ F ⇔ ϕ ∈ img(S∗ ).

The functor L(E, ) is not exact in general, since exactness at L(E, F + ) would mean
that for quotient mappins p : F  F + the adjoint p∗ : L(E, F ) → L(E, F + ) is onto,
i.e. every morphism ϕ : E → F + can be lifted along p : F  F + to a morphism
ϕ̃ : E → F .

4.110 Remark. Projective spaces and lifting of maps.

H`o o
Q
An (F) space E is called projective Fréchet space iff for every GO
quotient mapping Q : G  H of (F) spaces every T ∈ L(E, H)
T̃
has a lift T̃ ∈ L(E, G), i.e. Q ◦ T̃ = T . T
E

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Splitting sequences 4.112

Obviously every finite dimensional space is projective, since all linear mappings
on it are continuous. It was shown in [Gej78] that there are no other projective
Fréchet spaces.

4.111 Theorem. Splitting sequences (See [Vog87, 1.8 p.171]).

Let E and F be (F). Then
1. Let 0 → F → G → H → 0 be exact in (F).
Then 0 → L(E, F ) → L(E, G) → L(E, H) → 0 is exact.
⇔ 2. Let 0 → F → G → H → 0 be exact in (F).
Then any T ∈ L(E, H) lifts along G  H.
⇔ 3. Let 0 → F → G → E → 0 be exact in (F).
Then G → E has a right inverse.
⇔ 4. Let 0 → F → G → E → 0 be exact in (F). Then it splits,
inj pr
i.e. is isomorphic to the sequence 0 → F − 1→ F ⊕ E − 2→ E → 0
⇔ 5. Let 0 → F → G → E → 0 be exact in (F).
Then F → G has a left inverse.,
⇔ 6. Let 0 → H → G → E → 0 be exact in (F).
Then any T ∈ L(H, F ) extends along H  G.
⇔ 7. Let 0 → H → G → E → 0 be exact in (F).
Then 0 → L(E, F ) → L(G, F ) → L(H, F ) → 0 is exact.

Proof. ( 1 ⇔ 2 ) Since L(E, ) is left exact by 4.109 , 1 holds iff L(E, G) →

L(E, H) is onto, i.e. any T ∈ L(E, H) has a lift T̃ ∈ L(E, G).
( 2 ⇒ 3 ) By 2 the identity idE : E → E has a lift id
g E : E → G along G → E.

( 3 ⇒ 4 ) Let id = Q ◦ S : E → G → E. The isomorphism F ⊕ E → G is given by

(y, x) 7→ (y, S(x)) with inverse G → F ⊕ E, z 7→ (z − S(Q(z)), Q(z)).
( 4 ⇒ 2 ) Consider the pull-back

0 /F  /G //H /0
O O The bottom row is again an exact
pr1 T
sequence, hence splits by 4 , and
0 /F  inj1
/ G ×H E pr2
//E /0 thus gives a lifting
O
∼
= Φ T̃ := pr1 ◦Φ ◦ inj2 ,
0 /F  inj1
/ F ⊕E pr2
//E /0
h where Φ is the isomorphism.
inj2

( 7 ⇔ 6 ⇔ 5 ⇔ 4 ) is obtained by dualizing the arguments of ( 1 ⇔ 2 ⇔ 3 ⇔ 4 ).

4.112 Remark. Exactness of tensor-functors.

The algebraic tensor product functor ⊗ F is exact, since short exact sequences
of linear spaces are splitting and applying the functor ⊗ F to the splitting gives
splitting exact sequences.
The injective tensor product functor ⊗ε F preserves embeddings S, since we have by
definition natural embeddings E ⊗ε F ,→ L(E ∗ , F ) and L(E ∗ , ) obviously preserves
embeddings S : F1 ,→ F2 , in fact
(S∗ )−1 (NB,V ) = {T : (S ◦ T )(B) ⊆ V } = {T : T (B) ⊆ S −1 (V )} =: NB,S −1 (V ) .
It does not preserve quotient mappings, see [Kri07a, 4.29].

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4.114 Splitting sequences

The projective tensor product functor ⊗π F preserves quotient mappings Q : E1 →

E2 , since the image of U ⊗V ⊆ E1 ⊗π V under the linear map Q⊗F is the absolutely
convex hull of the image (⊗ ◦ (Q × F ))(U1 × V ) = ⊗(Q(U1 ) × V ) and hence is a
0-neighborhood in E2 ⊗π F .
The completion functor ( )∼ preserves short topologically exact sequences 0 → E ,→
F  G → 0, since Ẽ can be obtained as closure of E in F̃ . Thus E = Ẽ ∩ F , since
F̃
y ∈ Ẽ∩F = E ∩F ⇒ ∃xi ∈ E : xi → y ∈ F ⇒ 0 = Q(xi ) = Q̃(xi ) → Q̃(y) = Q(y),
i.e. y ∈ ker Q = E. Therefore, the mapping G ∼ = F/E = F/(Ẽ∩F ) → F̃ /Ẽ induced
from F ,→ F̃ is injective. It is an embeddings, since for every continuous seminorm
q in G, we can extend p := q ◦ Q to a continuous seminorm p̃ in F̃ which has to
vanish on the closure Ẽ of E in F̃ hence factors to a continuous seminorm q̃ on
F̃ /Ẽ, which induces q on F/E. Since F ,→ F̃ has dense image and F̃  F̃ /Ẽ is
onto the embedding F/E ,→ F̃ /Ẽ has dense image, and since F̃ /Ẽ is a Fréchet
space, it is the completion G̃ of G.
In order to describe the obstruction to exactness of the functor L we need injective
resolutions:

4.113 Proposition. Injective Resolution.

For every Fréchet space F there exists an injective resolution, i.e. a long exact
sequence 0 → F → I0 → I1 → I2 → · · · , where Ik is an injective Fréchet space for
each k ∈ N.

Proof. Let I0 be an injective Fréchet space into which F embeds by 4.108 .

Recursively, we may embed Ik / img(Ik−1 ) (where I−1 := F ) into an injective
Fréchet space Ik+1 and take as connecting map the composite of the quotient map
Ik  Ik / img(Ik−1 ) and this embedding Ik / img(Ik−1 ) ,→ Ik+1 .
Using injective resolutions we can construct the derived functors using homological
algebra:

4.114 Theorem. Derived functors.

There are functors Extk : lcsop × lcs → vs for k ∈ Z (called the right-derived
functors of L) and natural transformations δ such that:
1. Extk (E, F ) = 0 for k < 0.
2. Ext0 ∼= L.
3. Extk (E, F ) = 0 for all k > 0 if F is injective.
4. For every short exact sequence 0 → E − → E → E + → 0 there is a long
exact sequence
· · · → Extk (E + , F ) → Extk (E, F ) → Extk (E − , F ) −δ→ Extk+1 (E + , F ) → · · · .
For every short exact sequence 0 → F − → F → F + → 0 there is a long
exact sequence
· · · → Extk (E, F − ) → Extk (E, F ) → Extk (E, F + ) −δ→ Extk+1 (E, F − ) → · · · .
For fixed F the functor Ext∗R ( , F ) together with the natural transformation δ is up
to isomorphisms uniquely determined by 1 - 4 . And similarly for each fixed E.

Proof.
( 1 ) By 4.113 there is an injective resolution I of F :
0 → F → I0 → I1 → I2 → · · ·

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Splitting sequences 4.114

Applying L(E, ) to I (only!) gives a cochain complex

0 → L(E, I0 ) → L(E, I1 ) → L(E, I2 ) → · · ·

and we define Extk (E, F ) := H k (L(E, I)).

By [KriSS, 9.19] and [KriSS, 8.23] the linear spaces Extk (E, F ) are independent
on the injective resolution of F .
( 2 ) By definition Ext0R (E, F ) is just the kernel of L(E, I0 ) → L(E, I1 ) and by left
exactness in 4.109 the sequence 0 → L(E, F ) → L(E, I0 ) → L(E, I1 ) → · · · is
exact, hence this kernel is isomorphic to L(E, F ).
( 3 ) If F is injective then I0 := F and Ik := 0 for k > 0 gives an injective resolution.
Hence L(E, Ik ) = 0 and thus also Extk (E, F ) = H k (L(E, I)) = 0 for k > 0.
( 4 ) Let 0 ← E + ← E ← E − ← 0 be short exact and I be an injective resolution
of F . Since Ik is injective we have short exact sequences
0 → L(E + , Ik ) → L(E, Ik ) → L(E − , Ik ) → 0
and this gives a short exact sequence of cochain complexes since L is a bifunctor:
0 → L(E + , I) → L(E, I) → L(E − , I) → 0
By [KriSS, 7.30] we get a long exact sequence in (co)homology:

· · · → Extk (E + , F ) → Extk (E, F ) → Extk (E − , F ) −δ→ Extk+1 (E + , F ) → · · · .

Let 0 → F − → F → F + → 0 be short exact and I − and I + be corresponding

injective resolutions of F − and F + . We construct an injective resolution I of E and
an exact sequence of resolutions with

0 / E− /E / E+ /0

  
0 / I− /I / I+ /0

Take Ik := Ik− ⊕ Ik+ and put −

Ik−1 / / Ik−1 / / I+
inj1 pr2 k−1
dk := (d˜− , d+ ◦ pr ) : Ik → I −
k k 2 k+1
+
⊕ Ik+1 , d0k−1 dk−1 d00
k−1

where d˜k is an extension of dk : Ik →

− − −  |  
− Ik− / / Ik / / I+
Ik+1 along Ik− ,→ Ik . pr2 k
d˜−
This makes I into a chain complex and d−
k
k
dk d+
k
inj1 : I − → I and pr2 : I → I + into chain  |r pr1  
mappings. −
Ik+1 / / Ik+1 / / I+
inj1 pr2 k+1

Since Ik0 is injective, the sequences 0 → Ik− → Ik → Ik+ → 0 split and hence also
0 → L(E, Ik0 ) → L(E, Ik ) → L(E, Ik00 ) → 0 splits and, in particular, is exact. By
[KriSS, 7.30] we get a long exact sequence in (co)homology:

· · · → Extk (E, F − ) → Extk (E, F ) → Extk (E, F + ) −δ→ Extk+1 (E, F − ) → · · · .

(Uniqueness) We proceed by induction on k. For k ≤ 0 we have uniqueness by

1 and 2 . So we assume that we have two sequences of functors Ext∗ , which are
naturally isomorphic till order k, and we have natural connecting morphisms. Then
a diagram chase starting at a short exact sequence 0 → F − → F → F + → 0 with

andreas.kriegl@univie.ac.at c July 1, 2016 103

4.123 Splitting sequences

injective F shows that they are also isomorphic in order k + 1 on (E, F − ):

∼
... /0 / Extk (E, F + ) = / Extk+1 (E, F − ) /0 / ...

∼
= ∼
=
 
∼
... /0 / ExtkR (E, F + ) = / Extk+1 −
R (E, F )
/0 / ...

4.115 Proposition.
The statement Ext1 (E, F ) = 0 is equivalent to the equivalent conditions of 4.111 .

Proof. (⇒ 4.111.1 ) By 4.114.4 we have the exact sequence

0 → L(E, F ) → L(E, G) → L(E, H) → Ext1 (E, F ) → · · ·
| {z }
=0

(⇐ 4.111.1 ) Choose an injective I into which F embeds by 4.108 and consider

the short exact sequence 0 → F ,→ I  I/F → 0. Hence 0 → L(E, F ) →
L(E, I) → L(E, I/F ) → 0 is exact and in particular L(E, I) → L(E, I/F ) is onto.
Investigating the long exact sequence

0 → L(E, F ) → L(E, I)  L(E, I/F )  Ext1 (E, F ) → Ext1 (E, I) → · · ·

0
| {z }
=0
1
using 4.114.3 , gives Ext (E, F ) = 0.

For an additive description of (DN) and later of (Ω) similar to 4.88.2 for quasi-
normed spaces, we need the following:

4.119. Lemma. b b a
1 a+b a a+b
Let a, b > 0 and α, β ≥ 0. Then inf{ra α + rb
β : r > 0} = a (b) α a+b β a+b

Proof. Let f (r) := ra α + r1b β = ra+b α + β r−b . Then f 0 (r) = a α ra−1 − b rb+1
1


β
and hence f 0 (r) = 0 ⇔ ra+b α = a β. Thus
b

  − a+b b
   b
b bβ b b b  a − a+b
f (r) ≥ β+β = α a+b β 1− a+b 1 +
a aα a b
b
a a + b a a+b
b
 
= α a+b β a+b .
a b
Note that f (r) → +∞ for r & 0 if α > 0 and for r % +∞ if β > 0, hence the
infimum is attained if α, β > 0. Otherwise f (r) → 0 for r → 0 or r → +∞, hence
the statement is valid in this case as well.

4.123 Theorem. Characterization of the property (DN).

Let E be a Fréchet space with an increasing basis of semi-norms k kk corresponding
closed unit balls Uk and polars Bk := (Uk )o .
1. E is (DN) (see 3.13 ), i.e.
∃q ∀p ∃p0 ∃C > 0 : k k2p ≤ C k kq · k kp0 ;
⇔ 2. ∃q ∀p ∃p0 ∃C > 0 ∀r > 0 : k kp ≤ rk kq + C
r k kp ;
0

0 o o C o
⇔ 3. ∃q ∀p ∃p ∃C > 0 ∀r > 0 : Up ⊆ r Uq + r Up0 ;
⇔ 4. ∃q ∀ 0 < δ < 1 ∀p ∃p0 ∃C > 0 : k kp ≤ Ck k1−δ
q ·k kδp0 ;
0 1+d d
⇔ 5. ∃q ∃d > 0 ∀p ∃p ∃C > 0 : k kp ≤ Ck kq · k kp0 ;

104 andreas.kriegl@univie.ac.at c July 1, 2016

Splitting sequences 4.123

⇔ 6. There exists a log-convex basis of semi-norms ||| |||p on E, i.e.

2
∀p : ||| |||p ≤ ||| |||p−1 · ||| |||p+1 .

Note that in all these conditions we may assume w.l.o.g. that q < p < p0 , since for
p00 ≥ p0 we have k kp00 ≥ k kp0 and for p ≤ q we may take p0 = p and C = 1.
Note furthermore, that for k kp0 ≥ k kq only δ near 0 (and hence d = 1−δ δ near ∞)
are relevant, since for δ < δ 0 (and kykq 6= 0) we get
 kyk 0 δ  kyk 0 δ0 0 0
p p
kyk1−δ
q kykδp0 = kykq ≤ kykq = kyk1−δ
q kykδp0
kykq kykq
| {z }
≥1

Proof.
( 1 ⇔ 2 ) the minimum of r 7→ rkxkq + Cr kxkp0 is 2 Ckxkq kxkp0 by 4.119 .
p

Hence the inequality in 6 for all r > 0 is equivalent to k k2p ≤ 4Ck kq k kp0 .
C
( 2 ⇔ 3 ) From k kp ≤ rk kq + rk kp0 we get
 
1
r Uq ∩ r
C Up
0 ⊆ 2 Up and hence Upo ⊆ 2 r Uqo + C o
r Up0 .

Conversely,
Upo ⊆ r Uqo + C o
r Up0
implies that any u ∈ Upo can be written as u = r v + Cr u0 with v ∈ Uqo , u0 ∈ Upo0 , i.e.
C 0 C
|u(x)| ≤ r|v(x)| + r |u (x)| ≤ rkxkq + r kxkp
0

for all x ∈ E. Hence 2 holds, since kxkq = supu∈Uq |u(x)|.

( 1 ⇒ 6 ) Define a new basis of semi norms ||| |||k recursively by: ||| |||0 := k kq ;
2
∃p00 ∃C0 ≥ 1: ||| |||0 ≤ ||| |||0 · C0 k kp00 = ||| |||0 · ||| |||1 , where ||| |||1 := C0 k kp00 ;
2
∃p0k ∃Ck ≥ 1: ||| |||k ≤ ||| |||0 · Ck k kp0k ≤ ||| |||k−1 · ||| |||k+1 , with ||| |||k+1 := Ck k kp0k .
2
( 6 ⇒ 1 ) From ||| |||k ≤ ||| |||k−1 ||| |||k+1 we obtain that all ||| |||k are norms and
using |||x|||k /|||x|||k−1 ≤ |||x|||k+1 /|||x|||k for all x 6= 0, we get for all k ∈ N the
inequality
k 2k
|||x|||k Y |||x|||j Y |||x|||j |||x|||2k 2
= ≤ = , i.e. |||x|||k ≤ |||x|||0 |||x|||2k .
|||x|||0 j=1
|||x||| j−1 |||x||| j−1 |||x|||k
j=k+1

( 1 ⇒ 4 ) Put p0 := p and apply 1 iteratively to get pν+1 ≥ pν and a Cν with

k k2pν ≤ Cν k kq k kpν+1 .
Let 0 < δ < 1 and m ∈ N with 1
m+1 < δ. Since k kq is a norm, we have
m m−1 m−1 m−1
!
k kpν+1

k kp Y k kpν Y Y k k pm
≤ ≤ Cν ≤ Cν .
k kq ν=0
k kq ν=0
k kpν ν=0
k kp
Q 1
 m+1
m−1
If we put C := ν=0 Cν , we get
 1/(m+1)  δ
1− 1 1
k k pm k kpm
k kp ≤ Ck kq m+1 k km+1
pm = C k kq · ≤ C k kq ·
k kq k kq

1−δ
( 4 ⇒ 5 ) This follows directly with d := δ .

andreas.kriegl@univie.ac.at c July 1, 2016 105

4.127 Splitting sequences

( 5 ⇒ 1 ) We have ∀p ∃p0 , Cp : kxk1+d p ≤ Cp kxkdq kxkp0 and ∀p0 ∃p00 , Cp0 :

1+d
kxkp0 ≤ Cp0 kxkdq kxkp00 . Thus for d0 = 2d + d2 > 2d we get
0 2 0
kxkp1+d = kxkp(1+d) ≤ Cp1+d kxkd(1+d)
q Cp0 kxkdq kxkp00 = Cp0 Cp1+d kxkdq kxkp00
So we get 5 for some d ≥ 1 by induction and thus also for d = 1.

4.124 Definition. Generalized power series spaces (See [Vog85, p.256]).

For a set J and a : J → {t ∈ R : t ≥ 1} we consider the following generalized
power series spaces of infinite type:
K N
n o
λ∞
∞ (a) := f ∈ J
: kf k k := sup |f (t)| a(t)k
< ∞ for all k ∈ ,
t∈J

λ1∞ (a) := f ∈ KJ : kf kk := |f (t)| a(t)k < ∞ for all ∈ N .

n X o

t∈J

These Fréchet spaces λp∞ (a) are usally denoted Λp (J, a) for p ∈ {1, ∞} and for
p = 1 the index is often dropped.

4.125 Lemma (cf. [Vog77a, 2.4. Corollary p.115] ).

For each a ∈ RJ≥1 the space λ∞
∞ (a) has property (DN).

Proof. By definition kf kk := kf · ak k`∞ . Thus

kf k2p = kf · ap k2`∞ = kf 2 · a2p k`∞ ≤ kf k`∞ · kf · a2p k`∞ = kf k0 · kf k2p
Thus λ∞ 0
∞ (a) satisfies condition 4.123.1 for q := 0, p := 2p, and C := 1.

4.126 Lemma (See [Vog85, Example (3) p.256]).

Let I be a set and a : N × I → N be given by a(n, i) := n + 1.
ˆ ∼
Then `∞ (I)⊗s = λ∞ 1 ˆ ∼ 1
∞ (a) and ` (I)⊗s = λ∞ (a).

Proof. By 3.48 E 0 ⊗ε F ,→ L(E, F ). Since c0 (I) is Banach with dual c0 (I)∗ =

`1 (I) and `1 (I)∗ = `∞ (I) we get embeddings
`1 (I) ⊗ε s ,→ L(c0 (I), s) and `∞ (I) ⊗ε s ,→ L(`1 (I), s).
Let Pk : s → s be defined by Pk (x)n := xn for n < k and 0 otherwise. Then
Pk → id and for T ∈ L(`1 (I), s) we have Pk ◦ T → T with Pk ◦ T ∈ `1 (I)∗ ⊗ s. So
ˆ ∼
`∞ (I)⊗s ˆ ∼
= L(`1 (I), s) and analogously `1 (I)⊗s = L(c0 (I), s).
Furthermore, L(`1 (I), E) ∼ = `∞ (I, E) and L(c0 (I), E) ∼= `1 (I, E): In fact, T ∈
1 i
L(` (I), E) is uniquely determined by x := T (ei ) ∈ E for i ∈ I. Since T is
bounded, {xi : i ∈ I} ⊆ E has to be bounded. And conversely a bounded family
i 1
{x
P : i i∈ I} ⊆ E defines a continuous linear operator T : ` (I) → E, (yi )i∈I 7→
i yi x ∈ E. And the same arguments work also for the second isomorphism.
Finally note, that `∞ (I, s) ∼
= λ∞ 1 ∼ 1
∞ (a) and ` (I, s) = λ∞ (a): In fact, the seminorm
k kk : x 7→ sup{|(n + 1) xn | : n ∈ N} of s induces the seminorm of `∞ (I, s) by
k

taking the `∞ -Norm of (kxi kk )i∈I and corresponds to the seminorm k kk of λ∞ ∞ (a).
Replacing the supremum by the 1-Norm, gives the second isomorphism.

4.127 Proposition (See [Vog82, 1.1 p.540], [Vog85, Lemma 1.3 p.258], [Vog87,
4.3 p.185], and [Vog77a, Satz 1.5 p.111]).
Let F and G be Fréchet spaces and assume that G has property (DN ).
Then any exact sequence
Q
0 → λ∞
∞ (a) → F −→ G → 0
with a ∈ RJ≥1 splits.

106 andreas.kriegl@univie.ac.at c July 1, 2016

Splitting sequences 4.128

Proof. W.l.o.g. let E := λ∞∞ (a) ,→ F be the inclusion of a subspace. We have to

prove that it is complemented, i.e. there exists a left inverse φ : F → E to it.
Let evj ∈ E ∗ be given byevj (x) := x(j) for x ∈ E. Since |a(j)k evj (x)| =
|a(j)k x(j)| ≤ kxkk the set a(j)k evj : j ∈ J is equicontinuous for each k ∈ N.
By Hahn-Banach we can extend evj to ev e kj ∈ F ∗ for each k ∈ N such that
n o
e kj : j ∈ J is equicontinuous, thus contained in Uko for a suitable neigh-
a(j)k ev
bourhood Uk of 0 ∈ F . We can assume that Uk+1 ⊆ Uk for all k ∈ N.
Thus
 
e k+1
gjk := ev j e kj ∈
− ev 1
a(j)k
1
a(j)
o
Uk+1 +Uko ∩E o ⊆ 2
a(j)k
o
Uk+1 ∩E o =: 1
B
a(j)k k
⊆ Eo.
Since Q∗ : G∗ = ∼ (F/E)∗ ∼ = E o ⊆ F ∗ as cbs for the equicontinuous subsets by
[Kri07b, 7.4.4] and 4.72.2 and since G has property (DN ), there exists a bounded
set B ⊆ E o , which satisfies the conditions of 4.123.3 for a fixed fundamental
system of bounded sets Bk in E o . W.l.o.g. (by enlargeing Bk+1 ) we can assume
that
2−k−2
∀k ∈ N ∀r > 0 : Bk ⊆ rB + Bk+1
r
In particular, for r := a(j)2−k−1 we get by multiplication with 2a(j)−k
(1) 2a(j)−k Bk ⊆ a(j)−k+1 2−k B + a(j)−k−1 Bk+1 .
We now choose for fixed j recursively bkj with bkj ∈ a(j)−k Bk ⊆ E o :
Put b0j := 0. If bkj ∈ a(j)−k Bk is already chosen, we have gjk + bkj ∈ 2a(j)−k Bk .
Hence by ( 1 ) there exists a bk+1
j ∈ a(j)−k−1 Bk+1 such that
gjk + bkj − bk+1
j ∈ 2−k a(j)−k+1 B.
If we put
e kj − bkj ∈ F ∗ ,
φkj := ev
we get for k ≥ 1:
φk+1
j − φkj = gjk − bk+1
j + bkj ∈ 2−k a(j)−k+1 B ⊆ 2−k B.
Hence
∃φj := lim φkj ∈ F ∗ .
k→∞
Since φn+1
j = e n+1
ev j − bn+1
j ∈ 2a(j) −n o
Un+1 we have for k > n:
k−1
X
a(j)n φkj = a(j)n φn+1
j + a(j)n (φν+1
j − φνj ) ∈ 2Un+1
o
+ 2−n B.
ν=n+1

+ 2 B, i.e. a(j)n φj : j ∈ J is equicontinuous in F ∗ .

−n
n o

Thus a(j) φj ∈ 2Un+1
Therefore x → (φj (x))j∈J defines a continuous linear left inverse to E ,→ F , since
for x ∈ E we have
e kj (x) − bkj (x) = evj (x) − 0 = x(j).
φj (x) = lim φkj (x) = lim ev
k→∞ k→∞

In fact, it can be shown that the condition (DN) yields even a characterization:

4.128 Theorem [Vog87, 4.3 p.185].

Let supn ααn+1
n
< ∞, r ≤ +∞, and E a Fréchet space. Then
1. E is (DN);
⇔ 2. Ext1 (E, λ∞ ∞
r (α)) = 0, i.e. any ses 0 → λr (α) → G → E → 0 splits;
⇔ 3. If Q : G  H is a quotient mapping with kernel λ∞r (α)
then Q∗ : L(E, G) → L(E, H) is onto;

andreas.kriegl@univie.ac.at c July 1, 2016 107

4.131 Splitting sequences

⇔ 4. If S : H ,→ G is a closed embedding with quotient E

then S ∗ : L(G, λ∞ ∞
r ) → L(H, λr ) is onto.

Proof. ( 1 ⇒ 2 ) for r = +∞ is 4.127 .

( 1 ⇐ 2 ) is shown in [Vog87, 4.3 p.185].
( 2 ⇔ 3 ⇔ 4 ) is 4.111 and 4.115 .

4.129 Corollary [See64].

The restriction incl∗ : C ∞ (R, R)  C ∞ (R≥0 , R) has a continuous linear right
inverse.

Proof. We show first that the restriction map C[−2,2] ∞

(R)  C ∞ ([−1.1]) has
a continuous linear right inverse: By 1.16.4 C ∞ ([−1, 1]) = ∼ s and by 1.16.3
∞
C[−2,2] (R) =
∼ s. Moreover the kernel of the restriction map is the subspace

f ∈ C ∞ (R) : f (t) = 0 ∀|t| ≥ 2 and f (t) = 0 ∀|t| ≤ 1 =

n o

∞
= C[−2,−1] (R) ⊕ C[1,2]
∞
(R) ∼
=s⊕s∼
=s:
In fact s ∼
= s × s via (xk )k∈N 7→ ((x2k )k∈N , (x2k+1 )k∈N ): This mapping is obviously
linear and injective. It is continuous, since |(k + 1)q x2k | ≤ |(2k + 1)q x2k | and
|(k + 1)q x2k+1 | ≤ |(2k + 2)q x2k+1 |. It is onto s × s, since given y, z ∈ s the inverse
image is given by x2k := yk and x2k+1 := zk with
(
q |(2k + 1)q yk | ≤ |2q (k + 1)q yk | for n = 2k,
|(n + 1) xn | =
|(2k + 2)q zk | ≤ |2q (k + 1)q zk | for n = 2k + 1.
Thus we have a short exact sequence s ,→ s  s, which splits by 4.127 since s is
a power series space of infinite type by 1.15.4 and hence has property (DN) by
3.14.3 .
Using translation it suffices to consider the restriction map C ∞ (R) → C ∞ (R≥−1 ).
We choose a function ϕ ∈ C ∞ (R, [0, 1]) with ϕ(t) = 0 for all t ≥ 0 and ϕ(t) = 1 for
all t ≤ − 21 and decompose f ∈ C ∞ (R≥−1 ) as f = (1 − ϕ) · f + ϕ · f . Since (1 − ϕ) · f
is 0 on [−1, − 12 ] we can extend it by 0 to f˜0 ∈ C ∞ (R). By what we have shown
before the restriction of ϕ · f to [−1, 1] has an extension f˜1 ∈ C[−2,2]
∞
(R) ⊆ C ∞ (R).
Then f˜2 : t 7→ ϕ(t − 1 ) · f˜1 (t) is an extension of ϕ · f restricted to [−1, +∞), since
2
ϕ(t − 21 ) = 1 for all t with ϕ(t) 6= 0 and ϕ(t − 21 ) = 0 = ϕ(t) for all t ≥ 12 . Thus
f˜ := f˜0 + f˜2 is the desired extension of f , and it depends continuously and linearly
on f , since all intermediate steps do so.
More generally, it is shown in [Tid79, Folgerung 2.4 p.296] for compact K ⊆ Rn :
C ∞ (Rn )  E(K) has a continuous linear right inverse ⇔ E(K) (DN) ⇔ E(K) ∼ = s.
Here E(K) ∼ = C ∞ (Rn )/{f ∈ C ∞ (Rn ) : f |K = 0} denotes the Fréchet space of
Whitney jets on K.
Another application is:

4.130 Proposition. [Vog87, 7.1 p.193].

Let D := P (∂) be an elliptic linear PDO with constant coefficients on Rn with n ≥ 2
and U ⊆ Rn open and E a Fréchet space. Then
D∗ : L(E, C ∞ (U )) → L(E, C ∞ (U )) is onto ⇔ Ext1 (E, ker D) = 0 ⇔ E is (DN).

4.131 Corollary (See [Vog83, 6.1. Satz p.197], [Vog85, 2.6 p.260] ).
A Fréchet space F is (DN ) ⇔ ∃J ∃a ∈ RJ≥1 : F ,→ λ∞
∞ (a).

108 andreas.kriegl@univie.ac.at c July 1, 2016

Splitting sequences 4.132

Proof.
(⇒) Let J := k Bk for some basis of equicontinuous sets Bk ⊆ F ∗ . Then F can
S
N
be embedded into (`∞ (J)) in a natural way.
∞
By Borel’s theorem 0 → C[−1,0] (R) × C[0,1]
∞
(R) ,→ C[−1,1]
∞
 RN → 0 is exact and
∞
C[a,b] (R) ∼
= s by 1.16.3 . Moreover s ∼
= s × s via (xk )k∈N 7→ ((x2k )k∈N , (x2k+1 )k∈N )
by what we have shown in 4.129 . By tensoring this exact sequence of nuclear (F)
spaces with `∞ (J) (i.e. applying L(( )∗ , `∞ (J)) with the injective (F) space `∞ (J))
we get the (using 4.99 ) exact sequence of (F) spaces:

0 → s⊗` ˆ ∞ (J) → RN ⊗`
ˆ ∞ (J) → s⊗` ˆ ∞ (J) → 0.

∞ (a) by 4.126 , where a : N × J → N is given by (n, j) 7→ n + 1,

ˆ ∞ (J) ∼
Since s⊗` = λ∞
N
and R ⊗` (J) ∼
ˆ ∞
= (R(N) )∗ ⊗` = L(R(N) , `∞ (J)) ∼
ˆ ∞ (J) ∼ = `∞ (J)N , this sequence is

0 → λ∞ ∞ ∞ Q N
∞ (a) → λ∞ (a) −→ (` (J)) → 0.
N
Since F embeds into (`∞ (J)) we may consider the pullback(=preimage) Q−1 (F )
of F under Q, and get the short exact sequence
0 → λ∞
∞ (a) → Q
−1
(F ) → F → 0.

By 4.127 the sequence splits if F has property (DN ). We therefore get the
embedding F ,→ Q−1 (F ) ⊆ λ∞
∞ (a).


ˆ ∞ (J) 
/ s⊗` / s⊗` // RN ⊗` /0
Q
0 ˆ ∞ (J) ˆ ∞ (J)

/ λ∞  / λ∞ Q
/ / (`∞ (J))N /0
0 ∞ (a) ∞O (a) O
(1)
? s
(3)
?
(2)
0 / λ∞
∞ (a)
/ Q−1 (F ) //F /0

(⇐) Since λ∞
∞ (a) has property (DN ) by 4.125 , the converse follows from 3.14.2 .

Now we consider the dual situation.

4.132 Lemma. Characterization of the property (Ω).

Let k kk be an increasing basis of seminorms of a Fréchet space E, denote with
Uk := {x ∈ E : kxkk ≤ 1} the corresponding unit-balls and k k−k the Minkowski
functionals of Uko , i.e. kyk−k := kyk∗k := sup{|y(x)| : x ∈ Uk } = sup |y(x)|

kxkk : x ∈ E
for y ∈ E ∗ (cf. property (DN) in 4.123 ).

1. ∀p ∃p0 ∀k ∃C > 0 ∃ 0 < δ < 1 : k k−p0 ≤ C (k k−p )1−δ · (k k−k )δ ;

⇔ 2. ∀p ∃p0 ∀k ∃C > 0 ∃d > 0 : k k1+d d
−p0 ≤ C k k−k · k k−p
[Vog83, p.194]. [VW80, Korollar 2.2 p.232]. [Vog85, p.255].
0
⇔ 3. ∀p ∃p0 ∀k ∃k 0 ∃C > 0 ∀r > 0 : k k−p0 ≤ C rk k k−k + 1r k k−p
[VW80, Korollar 2.1 p.232];
0
⇔ 4. ∀p ∃p0 ∀k ∃k 0 ∃C > 0 ∀r > 0 : Up0 ⊆ C rk Uk + 1
r Up
[VW80, Definition 1.1 p.225];

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4.134 Splitting sequences

A Fréchet space E is said to be (Ω) iff these equivalent conditions are satisfied.
Note that we may assume that p0 > p and it suffices that k > p0 and d ∈ N, since
q ≥ p0 ⇔ k kq ≥ k k0p ⇔ Up0 ⊇ Uq ⇔ k k−p0 ≥ k k−q thus 1 holds for each p00 > p0
as well and 2 holds for each d0 > d as well.
Proof.
d
(1⇔2)δ= d+1 .
0 k0 +1
p
( 2 ⇔ 3 ) since the infimum of r 7→ αrk + β 1r is Ck0 αβ k0 by 4.119 .
0
( 3 ⇒ 4 ) Let k k−p0 ≤ Crk k k−k + 1r k k−p . Then
1 o r o
0 Uk ∩ U ⊆ Upo0
2Cr k 2 p
and by taking polars
 1 r o 0 2
Up0 ⊆ ((Up0 )o )o ⊆ k 0 U o
k ∩ Up ⊆ 3Crk Uk + Up .
2Cr 2 o r
0
( 3 ⇐ 4 ) Let Up0 ⊆ C rk Uk + 1r Up . Then every x ∈ Up0 can be written as as
x = Crn a + 1r b with a ∈ Uk and b ∈ Up . Thus for x∗ ∈ E ∗ we get
0 1
|x∗ (x)| ≤ Crk kx∗ k−k + kx∗ k−p
r
and taking the sup over x ∈ Up0 gives 3 .

4.133 Inheritance properties of (Ω)

(See [VW80, Satz 2.5 p.236], [MV92, 29.11 p.347]).

1. (Ω) is a topological invariant.

2. (Ω) is inherited by quotients.
3. λqr (α) has (Ω) for all r ≤ ∞ and 1 ≤ q < ∞.
4. λ1∞ (a) has (Ω) for all a ∈ RJ≥1 .

Proof. ( 1 ) is obvious in view of 4.132.4 .

( 2 ) Let F ,→ E be a closed subspace, π : E → E/F the canonical quotient
mapping, p a seminorm on E, and p̃ the corresponding norm on the quotient. Then
p̃<1 = π(p<1 ) thus applying π to 4.132.4 for the open unit balls of E gives the
same for E/F .
1 1
( 3 ) For λqr (α) let q + q0 = 1 and a(j) := eαj . Then
y y d y  y d
kyk−k · kykd−p = · = ·
ak `q0 ap `q0 ak `q0 ap `q0 /d
1+d
y  y d
 y
≥ k · p 0 +d/q 0 )
= k+p d q0 /(1+d)
a a ` 1/(1/q a `
y 1+d
1+d
= (k+p d)/(1+d) q0 = kyk−p0 ,
a `
k+p d k−p0
where k > p0 > p and d is the solution of p0 = 1+d , i.e. d = p0 −p .

( 4 ) follows by the same arguments as in 3 but for uncountable index sets J.

4.134 Theorem
(See [Vog77b, Theorem 2.3], [Vog85, Lemma 1.3 p.258], and [Vog87, 4.1 p.183]).

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Splitting sequences 4.134

Let E and F be Fréchet spaces and assume that E has property (Ω).
Then any exact sequence
Q
0 → E → F −→ λ1∞ (a) → 0
with a ∈ RJ≥1 splits.

Proof. We assume that E = ker Q is a subspace of F . Using 4.132.4 we find an

decreasing 0-nbhd basis in E of absolutely convex sets Uk and νk such that
1
(1) Uk−1 ⊆ rνk−1 Uk + Uk−2 for all r ≥ 2 and k ≥ 2.
r
Let (Wk )k∈N be a corresponding decreasing 0-nbhd basis in F with Wk ∩ E = Uk
and let ej denote the j-th unit vector in λ1∞ (a). For the canonical norms
X
kxkk := a(j)k |xj |
j

we have kej kk = a(j)k . By the open mapping theorem, Q(Wk ) ⊆ λ1∞ (a) is open.
Hence, for every k there exists an nk ∈ N and a Ck ≥ 1 with
ej 
n
∈ x : kxknk ≤ 1 ⊆ Ck Q(Wk ).
a(j) k

Thus there are dkj ∈ Ck a(j)nk Wk ∩ Q−1 (ek ) ⊆ F . We may assume that
nk+1 ≥ (1 + νk−1 )nk ≥ nk and Ck+1 ≥ 2kνk−1 (3Ck )1+νk−1 ≥ Ck
for all k ∈ N. Thus
 
dkj − dk−1
j ∈ C k a(j) nk
W k + C k−1 a(j)nk−1
W k−1 ∩ ker Q ⊆ 2Ck a(j)nk Uk−1

We claim that there are akj ∈ Ck+1 a(j)nk+1 Uk with

Rjk := dkj − akj ∈ Ck a(j)nk Wk + Ck+1 a(j)nk+1 Uk ⊆ 2Ck+1 a(j)nk+1 Wk .
Let a0j := 0 and assume ak−1
j is already constructed. Then
dkj − dk−1
j + ak−1
j ∈ 2Ck a(j)nk Uk−1 + Ck a(j)nk Uk−1 ⊆ 3Ck a(j)nk Uk−1 .
| {z }
=:ρ≥1

k
Multiplying 1 for r := ρ 2 with ρ gives the existence of
akj ∈ ρ rνk−1 Uk = (3Ck a(j)nk )1+νk−1 2k νk−1 Uk ⊆ Ck+1 a(j)nk+1 Uk
with
ρ
Rjk − Rjk−1 = (dkj − dk−1
j + ak−1
j ) − akj ∈ Uk−2 = 2−k Uk−2
r
Thus
X X
∃ Rj := lim Rjl = Rjk + (Rjl − Rjl−1 ) ∈ 2Ck+1 a(j)nk+1 Wk + 2−l Ul−2 ⊆
l→∞
l>k l>k
 
nk+1 −k nk+1


⊆ 2Ck+1 a(j) +2 Wk−1 ⊆ 1 + 2Ck+1 a(j) Wk−1 ⊆ F.
So we can define
X
R(x) := xj Rj ∈ F for all x = (xj )j∈N ∈ λ1∞ (a),
j

since
X Rj
R(x) = a(j)nk+1 xj ∈ kxknk+1 (1 + 2Ck+1 ) Wk−1 .
j
a(j)nk+1
Thus, for each k > 0,
pWk−1 (R(x)) ≤ (1 + 2Ck+1 ) kxknk+1 ,

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4.136 Splitting sequences

i.e. R ∈ L(λ1∞ (a), F ) and, since

Q(Rj ) = lim Q(Rjk ) = lim Q(dkj − akj ) = ej − 0,

k→∞ k→∞

we get Q ◦ R = id.

In fact, it has be shown that the condition (Ω) gives even a characterization:

4.135 Proposition [Vog87, 4.1 p.183].

Let supn ααn+1
n
< ∞ and F be a Fréchet space. Then

1. F is (Ω);
⇔ 2. Ext1 (λ1∞ (α), F ) = 0, i.e. any ses 0 → F → G → λ1∞ (α) → 0 splits.
⇔ 3. If Q : G  H is a quotient mapping with kernel F
then Q∗ : L(λ1∞ (α), G) → L(λ1∞ (α), H) is onto;
⇔ 4. If S : H ,→ G is a closed embedding with quotient λ1∞ (α)
then S ∗ : L(G, F ) → L(H, F ) is onto.

Proof. ( 1 ⇒ 2 ) is 4.134 .

( 1 ⇐ 2 ) is shown in [Vog87, 4.1 p.183].

( 2 ⇔ 3 ⇔ 4 ) is 4.111 and 4.115 .

And similar to 4.131 one obtains:

4.136 Corollary [Vog85, 3.2 p.263].

A Fréchet space F is (Ω) ⇔ F is a quotient of λ1∞ (a) for some a ∈ RJ≥1 .

Proof.
(⇒) We have the canonical resolution
Y Y
0→E→ Ek → Ek → 0.
k k
Q P
Let F := {x = (xk )k ∈ k Ek : kxk := kxk kk < ∞}, a Banach space which
contains each Ek as direct summand (and let Fk be a complement of Ek in F ).
Let {xi : i ∈ I} be a (w.l.o.g. infi-
nite) dense subset in F and 0 → 0O 0O 0O
K ,→ `1 (I)  F → 0 be the
resulting exact sequence. Taking
the tensor product with the ses 0 / F ⊗s
ˆ / F ⊗s
ˆ / FN /0
O O O
N
0 → s → s → K → 0 gives by
4.112 a diagram with exact rows
/ `1 (I)⊗s ˆ / `1 (I)⊗s
ˆ / `1 (I)N /0
and columns (since all factors are 0 O O O
Fréchet and always one of them is
nuclear). This gives a right exact 0 / K ⊗sˆ / K ⊗s
ˆ / KN /0
diagonal sequence O O O

(`1 (I)⊗s)⊕(K
ˆ ˆ → `1 (I)⊗s−
⊗s) ˆ Q→ F N → 0
0 0 0
and let N denote the kernel of Q.

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Splitting sequences 4.139

Taking the direct sum of the Q canonical 0O 0O

resolution
Q of E with 0 → 0 → k Fk →
k Fk → 0 gives the exact sequence: /E / FN / FN /0
0 O O
0 → E → FN → FN → 0
and by 4.133.2 also every quotient of /E /H / `1 (I)⊗s
ˆ /0
0 O
λ1∞ (a). Now take the pullback H to ob- O
tain the diagram on the right side.
Its second row splits by 4.134 (i.e. H ∼= NO NO
E ⊕ (`1 (I)⊗s))
ˆ and taking the pullback G
of its two columns gives:
0 0
Since N is the quotient of
(`1 (I)⊗s)
ˆ ⊕ (K ⊗s)ˆ ∼ = (`1 (I) ⊕ K)⊗s
ˆ
0O 0O 1 ˆ ∼
and hence of ` (I t K)⊗s = λ∞ (a)1

by 4.126 we have that N has

/N / E ⊕ (`1 (I)⊗s) / FN /0 property (Ω). Therefore the second
0 ˆ O
O row splits and the first column
shows that E is a quotient of
0 /N /G / `1 (I)⊗s
ˆ /0 G∼ = N ⊕ (`1 (I)⊗s).
ˆ Thus E is also
O O a quotient of (`1 (I) ⊕ K ⊕ `1 (I))⊗s.
ˆ
Since K also contains a dense
NO NO subset of cardinality ≤ |I| it is
a quotient of `1 (I) and since
`1 (I)3 ∼
= `1 (I t I t I) ∼= `1 (I) we
0 0 conclude that E is a quotient of
`1 (I)⊗s.
ˆ

For power series spaces λ10 (α) of finite type one needs the stronger condition (Ω):
4.137 Proposition. [Vog87, 4.2 p.184].
Let limn→∞ ααn+1
n
= 1 and E be a Fréchet space. Then
1. E has (Ω), i.e. ∀p ∃p0 ∀k ∀d > 0 ∃C > 0 : k k1+d d
−p0 ≤ C k k−k · k k−p (cf.
4.132.2 );
⇔ 2. Ext1 (λ10 (α), E) = 0, i.e. any ses 0 → E → G → λ10 (α) → 0 splits.
If all involved Fréchet spaces have a basis of Hilbert seminorms then 4.127 and
4.134 can be generalized to
4.138 Splitting theorem (See [MV92, 30.1 p.357], [Vog87, 5.1 p.186]).
Let 0 → E → G → F → 0 be a short exact sequence of (F) spaces having a basis of
Hilbert seminorms.
If E is (Ω) and F is (DN), then the sequence splits.

4.139 Universal linearizer.

These results can also be used for lifting problems of non-linear functions:
Let F(U, E) be a class of functions from some set U (e.g. an open subset of some
Kn ) into lcs E from a certain class.
The corresponding free space (or universal linearizer)
λ(U ) should be an lcs in this class with the following uni- U
δ / λ(U )
versal property:
∃! f˜∈L
There exists a δ ∈ F(U, λ(U )), such for every f ∈ F(U, E) ∀f ∈F
" 
there exists a unique f˜ ∈ L(λ(U ), E) with f˜ ◦ δ = f . ∀E

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4.139 Splitting sequences

Let us try to find λ(U ): For E := K we need a bijection δ ∗ : λ(U )∗ → F(U ) :=

F(U, K). Thus, if we have some reflexive lc-topology on F(U ) then λ(U ) =
F(U )∗ and δ ∗ should be the inverse of δF (U ) : F(U ) → F(U )∗∗ , i.e. f (t) =
δ ∗ (δF (U ) (f ))(t) = δF (U ) (f )(δ(t)) = δ(t)(f ) for all f ∈ F(U ) and all t ∈ U . So
δ : U → λ(U ) := F(U )∗ is the usual evaluation map.
We need that δ : U → F(U )∗ , x 7→ (f 7→ f (x)) belongs to F. Often it is the case,
that switching variables gives a bijection F(U, E 0 ) ∼= L(E, F(U )). For E := F(U ),
the map δ : U → E ∗ ,→ E 0 corresponds to id ∈ L(E, E), hence belongs to F(U, E 0 )
and, since it has values in E ∗ , it usually belongs even to F(U, E ∗ ).
Let now E be arbitrary. In order that δ ∗ : L(λ(U ), E) → F(U, E), T 7→ T ◦δ, makes
sense, we need that f ∈ F, T ∈ L ⇒ T ◦ f ∈ F, which is not a big limitation.
Is δ ∗ injective? So let T ∈ L(λ(U ), E) be such that f := T ◦ δ = 0, hence 0 =
x∗ ◦ T ◦ δ = δ ∗ (x∗ ◦ T ) : U → K for all x∗ ∈ E ∗ . Since δ ∗ : λ(E)∗ → F(U ) is the
inverse of δ : F(U ) → F(U )∗∗ it follows that x∗ ◦ T = 0, and consequently T = 0.
Note, that we can deduce that the image of δ : U → λ(U ) generates a dense linear
subspace, since every continuous linear functional T on λ(U ) which vanishes on the
image of δ, i.e. δ ∗ (T ) = 0, has to be 0.
Is δ ∗ onto? So let f ∈ F(U, E) and consider f ∗ : E ∗ → F(U ), x∗ 7→ x∗ ◦ f . This
is well-defined by the assumption above. In order to show that it is bounded, we
consider the associated mapping ff∗ : U → (E ∗ )0 , which is just f : U → E ,→ (E ∗ )0
and belongs to F. So f ∗ is bounded and hence f ∗∗ : λ(U ) = F(U )0 → (E ∗ )0 is
continuous. Since f ∗∗ ◦ δ = δ ◦ f : U → E → (E ∗ )0 its values on the image of δ lie
in E and, since this image generates a dense subspace, f ∗∗ is the required inverse
image for complete E.
Thus we have shown:

Proposition.
Let F(U, E) be function spaces with the following properties:
f ∈ F, T ∈ L ⇒ T ◦ f ∈ F.
1.
If ι : G ,→ E is a closed embedding, then f ∈ F(U, G) ⇔ ι ◦ f ∈ F(U, E).
2.
F(U, E 0 ) ∼
3. = L(E, F(U )) by switch of variables.
F(U ) carries a reflexive lc-topology.
4.
Let λ(U ) := F(U )∗ then δ ∗ : L(λ(U ), E) ∼
= F(U, E) is a linear bijection for each
complete lcs E with complete dual E ∗ .

Examples.
( 1 ) T ∈ L, f ∈ F ⇒ T ◦ f ∈ F:
For `∞ , C ∞ (See [KM97, 2.11 p.24]), H ([KN85, 2.6 p.283]), and C ω ([KM90,
1.9 p.10]) this is easily checked.

( 2 ) ι ◦ f ∈ F ⇒ f ∈ F:
For `∞ , C ∞ , H, and C ω this is obvious since these mappings can be tested by the
continuous linear functionals.
( 3 ) F(U, E 0 ) ∼
= L(E, F(U )):
For C ∞ see [FK88, 4.4.5], for H see [KN85, 2.14 p.288], for C ω see [KM90, 6.3.3
p.37], and for `∞ see [Kri07a, 4.7.4].
( 4 ) F(U ) reflexive:
= sN ).
C ∞ (U ) is nuclear (F) and has (Ω), but not (DN) (∼

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Splitting sequences 4.140

=C
H(U ) is nuclear (F) and a power series space, it has always (Ω) and only for U ∼
(DN).
C ω (U ) is complete ultrabornological (N) and its dual is complete nuclear (LF).
`∞ : For bornological spaces X one has `∞ (X) = (`1 (X))∗ by [FK88, 5.1.25] and
`1 (X) = (c0 (X))∗ by [FK88, 5.1.19], where
`1 (X) := {f ∈ RX : carr(f ) is bounded and kf k1 < ∞} and
c0 (X) := {f ∈ RX : carr(f ) countable and ∀B ∀ε > 0 : {x : |f (x)| > ε} finite.}.
However, λ(X) = `1 (X) for F := `∞ by [Kri07a, 4.7.4].
In many situations one can show better density conditions for the image of δ (like
Mackey-denseness) and hence gets the universal property for spaces E being less
complete (like Mackey-complete).
For U ⊆ Rn is open, it has been show in [FK88, 5.1.8] that λ(U ) = C ∞ (U, R)∗ is
universal for C ∞ -mappings into Mackey-complete spaces. For open U ⊆ Cn , it has
been shown in [Sie95] that λ(U ) = H(U )∗ is universal for H-mappings into Mackey-
complete spaces. The free convenient vector space for real-analytic mappings has
been considered in [KM90] and for sequentially complete spaces in [BD01]. In
[FK88, 5.1.24] it is shown that λ(X) = `1 (X) is universal for `∞ -mappings into
Mackey-complete spaces.

4.140 Parameter dependance of PDO solutions.

Particular cases for surjective PDO’s D := P (∂) : G(W )  G(W ) have been con-
sidered and (F-)parameter dependence of the solutions discussed: Let Ei := G(Wi )
and D : E1  E2 be onto. Is D∗ : F(U, E1 ) → F(U, E2 ) onto? Using the universal
linearizer λ(U ) for the function space F(U, ), this question is reduced to the sur-
jectivity of D∗ : L(λ(U ), E1 ) → L(λ(U ), E2 ). Using the suggested isomorphism one
obtains under appropriate conditions the following descriptions for the extension of
D : G(W1 ) → G(W2 ):
∗ ˆ ∗
F(U, G(W1 )) ∼
= L(F(U ) , G(W1 )) ∼
= F(U )⊗G(W ∼ ∼
1 ) = L(G(W1 ) , F(U )) = G(W1 , F(U ))
D∗ D∗ F (U )⊗D D ∗∗ D̃
    
∗ ˆ ∗
F(U, G(W2 )) ∼
= L(F(U ) , G(W2 )) ∼
= F(U )⊗G(W 2 ) = L(G(W2 ) , F(U )) = G(W2 , F(U ))
∼ ∼

[BD98, Corollary 39 p.34] If D : C ω (R)  C ω (R) onto then one can find solutions
depending holomorphically on a parameter in C. By [BD01, Proposition 9 p.501]
for every elliptic surjective linear PDO D := P (∂) : C ω (U ) → C ω (U ) with constant
coefficients and open U ⊆ Rn the extension D ⊗ E : C ω (U, E) → C ω (U, E) is
surjective if E is (F) or the strong dual of a (F)-space with (DN).
In contrast, by [BD01, Theorem 8 p.501] for every elliptic surjective linear PDO
D := P (∂) : C ω (R2 ) → C ω (R2 ) with constant coefficients the extension D ⊗ H(D̄) :
C ω (R2 , H(D̄)) → C ω (R2 , H(D̄)) is not surjective.
[BD01, Theorem 6 p.499] and [BD98, Theorem 38 p.33]: For open sets Ui ⊆ Rni
let T : C ω (U1 ) → C ω (U2 ) be a continuous linear surjective mapping. Then T ⊗ E :
C ω (U1 , E) → C ω (U2 , E) is onto provided E is (F)+(DN) or (E is complete+(LB)
and E ∗ is (Ω)) or E is a (F)-quojection, i.e. every quotient with a continuous norm
is a Banach space.
For (sequentially) complete lcs E and open U ⊆ Rn one has a linear bijection
C ω (U, E) ∼
= C ω (U )εE = L(C ω (U )∗β , E) ∼
= L((E ∗ , τc ), C ω (U )) by [BD01, Theorem
2 p.496]

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4.142 Splitting sequences

Locally bounded linear mappings

In this section, we describe situations where continuous linear mappings are even
locally bounded. If the domain space is a power series space of finite type, then
the characterizing property for the range space is (DN). And if the range space is
such a power series space, then the characterizing property of the domain space
is (Ω). For power series spaces of infinite type, the characterizing properties for
the other involved space are (LB∞ ) and (LB ∞ ). We give applications to vector
valued real-analytic mappings and mention applications to holomorphic functions
on Fréchet spaces.

4.141 Definition and Remark. Locally bounded operators.

A linear map T : E → F between lcs is called locally bounded if there exists a
0-nbhd U with T (U ) bounded. We will denote by LB(E, F ) the space of all locally
bounded linear maps from E to F .
We have LB(E, F ) ⊆ L(E, F ): Let U ⊆ E be a 0-nbhd with T (U ) bounded and
V ⊆ F be an arbitrary 0-nbhd. Then ∃C > 0: T (U ) ⊆ C V and hence C1 U ⊆
T −1 (V ), i.e. T is continuous.
We are interested in pairs (E, F ) for which LB = L.
If E or F is a normed space, then LB(E, F ) = L(E, F ): Let T ∈ L(E, F ). If E
is normed, than T (U ) is bounded for the unit ball U := oE. If F is normed, than
U := T −1 (oF ) is a 0-nbhd with T (U ) ⊆ oF bounded.
Note that idE ∈ LB(E, E) ⇔ E is normable, since U = id(U ) is a bounded 0-nbhd.
If Q : E  E1 is a quotient mapping and S : F1 ,→ F and is an embedding
then LB(E, F ) = L(E, F ) ⇒ LB(E1 , F1 ) = L(E1 , F1 ): For T ∈ L(E1 , F1 ) we have
that S ◦ T ◦ Q ∈ L(E, F ) = LB(E, F ), hence there exists a 0-nbhd U ⊆ E with
S(T (Q(U ))) ⊆ F bounded. Since Q is open, the set U1 := Q(U ) ⊆ E1 is a 0-nbhd
and since S is an embedding T (U1 ) = T (Q(U )) is bounded.
Let LB(E, F ) = L(E, F ). If there is an embedding E ,→ F , then E is normed,
since then LB(E, E) = L(E, E). And if there is a quotient mapping E  F , then
F is normed, since then LB(F, F ) = L(F, F ).
If E is a Fréchet space and LB(E, KN ) = L(E, KN ), then E is normable: If E is
not normable, then there exists a quotient mapping Q ∈ L(E, RN ) by 4.77 , hence
KN would have to normable but is not.
4.142 Proposition [BD98, Theorem 16 p.22], [BD01, Theorem 2 p.496].
The bijection δ ∗ : L(C ω (U )∗ , E) → C ω (U, E) from 4.139 for open U ⊆ Rn and
Fréchet spaces E maps LB(C ω (U )∗ , E) onto Ctω (U, E), the space of topologically
real-analytic mappings, i.e. mappings which are locally representable by a convergent
power series.
δ∗
L(C ω (U )∗ , E) / / / C ω (U, E)
O O

? ?
LB(C ω (U )∗ , E) / / / Ctω (U, E)

Sketch of proof. It is easy to see that f ∈ Ctω (R, E) is locally C ω into some EB
and by 4.90.1 even globally, hence corresponds to an element in L(C ω (R)∗ , EB ) =
LB(C ω (R)∗ , EB ) ⊆ LB(C ω (R)∗ , E).

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Locally bounded linear mappings 4.144

Conversely, T ∈ LB(C ω (R)∗ , E) ⇒ ∃B : T ∈ LB(C ω (R)∗ , EB ). Thus δ ∗ (T ) ∈

C ω (R, EB ) ⊆ Ctω (R, E), by a Baire argument, see [KM90, 1.6 p.8].
Thus, in order to get C ω (R, E) = Ctω (R, E) we have determine whether L = LB?

4.143 Definition.
Let E and F be (F) with increasing bases of seminorms (k kk )k∈N and (k kn )n∈N .
For linear T : E → F consider
kT kk,n := sup kT xkn ∈ [0, +∞].
kxkk ≤1

Note that kT kk+1,n ≤ kT kk,n ≤ kT kk,n+1 and

(1) T ∈ L(E, F ) ⇔ ∀n ∈ N ∃kn ∈ N : kT kkn ,n < ∞
(2) T ∈ LB(E, F ) ⇔ ∃k 0 ∈ N ∀n ∈ N : kT kk0 ,n < ∞

Proof.
( 1 ) T ∈ L(E, F ) ⇔ ∀n ∈ N ∃kn ∈ N ∃C > 0 : kT (x)kn ≤ Ckxkkn .
( 2 ) T ∈ LB(E, F ) ⇔ ∃k 0 ∈ N ∀n ∈ N ∃C > 0 ∀x : kxkk0 ≤ 1 ⇒ kT (x)kn ≤ C.

4.144 Lemma. Characterizing L = LB (See [Vog83, 1.1 p.183]).

Let E and F be (F) with increasing bases of seminorms (k kk )k∈N and (k kn )n∈N .
Then
1. L(E, F ) = LB(E, F );
⇔ 2. ∀k ∈ NN ∃k 0 ∀n ∃n0 ∃C > 0 ∀T ∈ L(E, F ) : kT kk0 ,n ≤ C maxm≤n0 kT kkm ,m .

W.l.o.g. k % ∞, since validity of 2 for k implies it for any k ≤ k.

Proof. For k ∈ NN consider
Gk := T ∈ L(E, F ) : kT kkn ,n < ∞ for all n ∈ N ,
n o

a Fréchet space with respect to the seminorms k kkn ,n for n ∈ N.

For each k 0 ∈ N let
Hk0 := T ∈ L(E, F ) : kT kk0 ,n < ∞ for all n ∈ N ,
n o

a Fréchet spaces with respect to the seminorms k kk0 ,n for n ∈ N.

Since {x : kxkk0 +1 ≤ 1} ⊆ {x : kxkk0 ≤ 1} we have kT kk0 +1,n ≤ kT kk0 ,n and thus
S
continuous inclusions
S Hk0 ⊆ Hk0 +1 . By 4.143.1 L(E, F ) = k Gk and by 4.143.2
LB(E, F ) = k0 Hk0 .
S
( 1 ⇒ 2 ) By 1 we have Gk ⊆ L(E, F ) = LB(E, F ) = k0 Hk0 Since the inclu-
sions Gk ⊆ L(E, F ) and Hk0 ⊆ L(E, F ) are continuous (for B ⊆ C Uk0 we have
sup{kT (x)kn : x ∈ B} ≤ C kT kk0 ,n ) we can apply Grothendieck’s Factorization
Theorem 2.6 to obtain a k 0 ∈ N such that Gk ⊆ Hk0 and the inclusion is continu-
ous, i.e.
∀n ∈ N ∃n0 ∈ N ∃C > 0 : kT kk0 ,n ≤ C max0 kT kkm ,m .
m≤n

1 ⇐ 2 ) Let T ∈ L(E, F ). By 4.143.1 ∃k ∈ NN : T ∈ Gk and by 2 :

∃k 0 ∀n ∃n0 ∃C > 0 : kT kk0 ,n ≤ C max0 kT kkm ,m .
m≤n

Hence
kT kk0 ,n < ∞ for all n,
i.e. T ∈ Hk0 ⊆ LB(E, F ).

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4.146 Locally bounded linear mappings

4.145 Lemma (See [Vog83, 1.3 p.184]).

Let B = {b(k) : k ∈ N} be a Köthe matrix and F a Fréchet space with increasing
basis of seminorms k kk . Then
1. L(λ1 (B), F ) = LB(λ1 (B), F );
⇔ 2. ∀k ∈ NN ∃k 0 ∀n ∃n0 ∃C > 0 ∀j ∀y ∈ F : kykn
(k0 ) ≤ C maxm≤n0 kykm
(k ) .
bj bj m

W.l.o.g. k % ∞.
Proof.
( 1 ⇒ 2 ) follows from 4.144 for T := prj ⊗y with y ∈ F and prj (x) := xj for
x ∈ λ1 (B) =: E, since
n o kyk
n
kT kk,n = sup kT xkn : kxkk ≤ 1 = sup |xj | kykn : kx · b(k) k`1 ≤ 1 = (k) .

bj

( 1 ⇐ 2 ) Since ej is an (absolute) Schauder-basis of E := λ1 (B) by 1.21 every

T ∈ L(E, F ) is of the form
X  X
T (x) = T prj (x) ej = prj (x) yj , where yj := T (ej ),
j j
X (k )0 kyj kn kyj kn
kT (x)kn ≤ bj | prj (x)| · sup = kxkk0 · sup ,
j∈N j∈N
(k0 ) (k0 )
j∈N bj bj
(km )
and ∀m ∃km ∃Cm > 0 : kyj km = kT (ej )km ≤ Cm kej kkm = Cm bj .

By 2 we have ∃k 0 ∀n ∃n0 ∃C > 0:

kyj kn  kyj km 
kT kk0 ,n ≤ sup (k0 ) ≤ sup C max0 (k ) ≤ C max0 Cm < ∞,
j∈N b j∈N m≤n b m m≤n
j j

i.e. T ∈ LB(E, F ) by 4.143.2 .

4.146 Theorem (See [Vog83, 2.1 p.186]).

Let β := (βj )j∈N be a shift-stable sequence, i.e. supn ββn+1 n
< ∞, and F a
Fréchet space with increasing basis of seminorms k kk . Then
1. L(λ10 (β), F ) = LB(λ10 (β), F );
⇔ 2. F has property (DN ) (See 4.123 ).
( 1 ⇐ 2 ) is valid without the assumption on β.
= K ⊕λpr (β), via Φ : x 7→ (x0 , S(x)), where
The shift-stability is equivalent to λpr (β) ∼
S(x)j := xj+1 .
Proof. By 1.26.1 we may replace λ10 (β) by the isomorphic space λ11 (β). Let
(k)
0 < ρk % 1 for k → ∞, i.e. bj := eρk βj describes the Köthe-matrix B for
λ11 (β) := λ(B).
( 1 ⇐ 2 ) Let k ∈ NN . By 4.123.5 the property (DN) means:
∃q ∃d > 0 ∀p ∃p0 ≥ q ∃C ≥ 1 : k k1+d
p ≤ C k kdq k kp0 .
(Note, that because of 4.131 , it would be enough to consider F = λ∞
∞ (a) and
0
hence q = 0, d = 1, p = 2p, and C = 1 by 4.125 .)
Now choose a k 0 > kq such that
1 − ρk 0
& 0 for k 0 → ∞ .


d>
ρk0 − ρkq

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Locally bounded linear mappings 4.147

Let y ∈ F and j ∈ N be fixed.

If
kykq < e(ρkq −ρk0 )βj kykp ,
then
kyk1+d
p ≤ C kykdq kykp0 ≤ C ed(ρkq −ρk0 )βj kykdp kykp0 .
By hypothesis d(ρkq − ρk0 ) ≤ ρk0 − 1 ≤ ρk0 − ρkp0 , so we get
(ρk0 −ρk )βj
kykp ≤ C e p0 kykp0 .
Otherwise,
kykp ≤ e(ρk0 −ρkq )βj kykq .
In any case
kykp n kyk
q kykp0 o kykm
≤ max ρ β
, C ρk 0 βj
≤ C max0 ρk βj
eρk0 βj e kq j
e p m≤p e m

and 4.145.2 gives 2 .

( 1 ⇒ 2 ) By 4.145.2 for the sequence k := id ∈ NN we have

∃k 0 ∀n ∃n0 ∃C > 0 ∀y ∈ F ∀j : kykn e−ρk0 βj ≤ C max0 kykm e−ρm βj .

m≤n

W.l.o.g. n0 ≥ max{n, k 0 + 1} and thus

kykn e−ρk0 βj ≤ C max0 kykm e−ρm βj ≤ C max kykk0 , kykn0 e−ρk0 +1 βj , since

m≤n
(
kykk0 · 1 for m ≤ k 0 ,
kykm e−ρm βj ≤
kykn0 e−ρk0 +1 βj for k 0 < m ≤ n0 .

Let b := supn βn+1

βn < ∞ and take y ∈ F . If there exists a j ∈ N such that

kykn e−ρk0 βj+1 ≤ Ckykk0 < kykn e−ρk0 βj (& 0 for j → ∞),
then
kykn ≤ eρk0 βj C max kykk0 , kykn0 e−ρk0 +1 βj = C kykn0 e(ρk0 −ρk0 +1 )βj

ρ 0 −ρ 0  kyk 0 d
−ρ 0 βj+1 k b+1 k
k
≤ Ckykn0 e k ρ 0
k ≤ Ckykn0 C ,
kykn
ρ 0 −ρ 0
where d := k b+1ρ 0 k , i.e. kyk1+d
n ≤ C 1+d kykn0 kykdk0 .
k
−ρk0 β0
If no such j exists, then e kykn ≤ Ckykk0 and we get
kyk1+d
n ≤ kykn0 kykdn ≤ kykn0 (C eρk0 β0 )d kykdk0 .
Hence in both cases
kyk1+d
n ≤ C 0 kykn0 kykdk0 with C 0 := C d max{C, eρk0 β0 d },
which is equivalent to (DN) by 4.123.5 with q := k 0 , p := n, and p0 := n0 .

4.147 Proposition (See [Vog83, 1.4 p.185]).

Let A = {a(k) ∈ RJ+ : k ∈ N} be a Köthe matrix, E a Fréchet space with decreasing
0-nbhd basis {Uk : k ∈ N} and Minkowski-functionals k k−k of the polars Uko . Then
1. L(E, λ∞ (A)) = LB(E, λ∞ (A));
⇔ 2. ∀k ∈ NN ∃k 0 ∀n ∃n0 ∃C > 0 ∀j ∀x∗ : aj kx∗ k−k0 ≤ C maxm≤n0 aj
(n) (m)
kx∗ k−km .

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4.148 Locally bounded linear mappings

Proof. This proof is similar to the proof of 4.145 .

( 1 ⇒ 2 ) follows from 4.144 for T := x∗ ⊗ ej with j ∈ N and x∗ ∈ E ∗ .
( 1 ⇐ 2 ) Let T ∈ L(E, λ∞ (A)) and put x∗j := prj ◦T ∈ E ∗ . Then
(n) (n)
kT (x)kn ≤ sup |aj x∗j (x)| ≤ kxkk0 sup aj kx∗j k−k0
j j∈N
(m)
sup aj kx∗j k−km ,


≤ kxkk0 C max0
m≤n j∈N
| {z }
=:kT kkm ,m

by 2 . This implies
kT kk0 ,n ≤ C max0 kT kkm ,m ,
m≤n
∞
i.e. T ∈ LB(E, λ (A)) by 4.144 .

4.148 Theorem. [Vog83, 4.2 Satz p.190].

Let (αj )j∈N be a shift-stable sequence and E a Fréchet space. Then
1. L(E, λ∞ ∞
0 (α)) = LB(E, λ0 (α));

⇔ 2. E has property (Ω) (see 4.137 ).

( 1 ⇐ 2 ) is valid without the assumption on α.

Proof. By 1.26.1 we may replace λ∞ ∞

0 (α) by the isomorphic space λ1 (α). Let
(k) ρk αj
0 < ρk % 1 for k → ∞, i.e. aj := e describes the Köthe-matrix A for
λ1 (α) := λ (A). W.l.o.g. we may assume that limk→∞ ρk1−ρ
∞ ∞
−ρk−1 = 0, e.g. take
k

ρk := 1 − k! . Let {Uk : k ∈ N} be an increasing 0-nbhd basis of E and k k−k the

1

Minkowski-functional of Uko ⊆ E ∗ .
( 1 ⇐ 2 ) Let k ∈ NN be given. For p := k0 choose p0 according to (Ω), i.e.
∀n ∀d > 0 ∃C ≥ 1 : k k1+d d
−p0 ≤ C k k−n · k k−p

For every n ∈ N let n0 ≥ p with ρn0 > ρn and d > 0 with d(ρn − ρ0 ) ≤ ρn0 − ρn .
Thus there exists a C ≥ 1 such that
k k1+d d
−p0 ≤ C k k−kn0 · k k−k0 .

For x∗ ∈ E ∗ and j ∈ N either eρn αj kx∗ k−p0 < eρ0 αj kx∗ k−k0 or
kx∗ k1+d ∗ ∗ d ∗
−p0 ≤ C kx k−kn0 · kx k−k0 ≤ C kx k−kn0 e
d(ρn −ρ0 )αj
kx∗ kd−p0 ,
i.e. kx∗ k−p0 ≤ C e(ρn0 −ρn )αj kx∗ k−kn0 .
In both cases we have 4.147.2
eρn αj kx∗ k−p0 ≤ max eρ0 αj kx∗ k−k0 , C eρn0 αj kx∗ k−kn0 ≤ C max0 eρm αj kx∗ k−km ,

m≤n

with k 0 := p0 , hence L = LB by 4.147 .

( 1 ⇒ 2 ) Let p ∈ N and consider the sequence k : n 7→ p + n. By 4.147.2
∃p0 ∀n ∃n0 ∃Cn ≥ 1 ∀x∗ ∈ E ∗ ∀j :
eρn αj kx∗ k−p0 ≤ Cn max0 eρm αj kx∗ k−km
m≤n
n o
≤ Cn max eρn−1 αj kx∗ k−p , eαj kx∗ k−kn ,

120 andreas.kriegl@univie.ac.at c July 1, 2016

Locally bounded linear mappings 4.149

since (
eρn−1 αj kx∗ k−k0 for m < n,
eρm αj kx∗ k−km ≤
eαj kx∗ k−kn for n ≤ m ≤ n0 .
Let x∗ ∈ E ∗ . If there exists a j ∈ N such that
e(ρn −ρn−1 )αj−1 kx∗ k−p0 ≤ Cn kx∗ k−p < e(ρn −ρn−1 )αj kx∗ k−p0 (% 0 for j → ∞),
(1−ρn )αj
then, since dn := supj (ρn −ρn−1 )αj−1 ,
n o
kx∗ k−p0 ≤ e−ρn αj Cn max eρn−1 αj kx∗ k−p , eαj kx∗ k−kn = Cn e(1−ρn )αj kx∗ k−kn
 kx∗ k−p dn ∗
≤ Cn e(ρn −ρn−1 )αj−1 dn kx∗ k−kn ≤ Cn Cn ∗ kx k−kn ,
kx k−p0
1+dn
i.e. kx∗ k−p0 ≤ Cn1+dn kx∗ k−kn kx∗ kd−p
n
.
If no such j exists, then Cn kx k−p < e(ρn −ρn−1 )α0 kx∗ k−p0 . Hence
∗
n o
kx∗ k−p0 ≤ e−ρn αj Cn max eρn−1 αj kx∗ k−p , eαj kx∗ k−kn = Cn e(1−ρn )α0 kx∗ k−kn
and thus we obtain in both cases
kx∗ k1+d 0 ∗ ∗ dn 0 dn (1−ρn )α0
−p0 ≤ Cn kx k−kn · kx k−p , where Cn := Cn max{Cn , e
n
}.
Since kn → +∞ and dn → 0, condition
(Ω) ∀p ∃p0 ∀k ∀d > 0 ∃C ≥ 1 : k k1+d d
−p0 ≤ C k k−k · k k−p
follows.
For power series spaces λp∞ (α) of infinite type one needs new (smaller) classes:

4.149 Theorem. [Vog83, 3.2 Satz p.188].

Let (βj )j∈N be a shift-stable sequence and F be a Fréchet space with increasing basis
of seminorms k kk . Then
1. L(λ1∞ (β), F ) = LB(λ1∞ (β), F );
⇔ 2. F has property (LB∞ ), i.e.
∀ρ ∈ RN 0 0 0 1+ρm
+ ∃k ∀n ∃n ∃C > 0 ∀y ∃m ∈ [n, n ] : kykn ≤ C kykρkm
0 kykm .

( 1 ⇐ 2 ) is valid without the assumption on β.

Similary as in 4.123 we may assume k 0 ≤ n ≤ n0 and ρ % ∞:

 ρ ρ0
kykk0 m

kykn kykk0 m
≤C ≤C for each ρ0 ≤ ρ.
kykm kykn kykn

Obviously (use 4.123.5 and ρ := constd ) one has: (LB∞ ) ⇒ (DN).

In fact, recall:
(DN) ∃k 0 ∃d > 0 ∀n ∃n0 ∃C > 0 : k k1+d
n ≤ C k kdk0 k kn0 ;

Proof.
( 1 ⇐ 2 ) Let k ∈ NN with k % +∞ be arbitrary. By 2 we have for ρ := k
∃k 0 ∀n ∃n0 ≥ k 0 ∃C > 0 ∀y ∃m ∈ [n, n0 ] : kyk1+k
n
m
≤ C kykkkm
0 kykm .

Put k 00 := kk0 + 1. For given j either

00
kykn e−k βj
≤ kykk0 e−kk0 βj
or
00
(kk0 −k
kyk1+k
n
m
≤ C kykkkm
0 kykm ≤ C e
)βj km
kykknm kykm

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4.150 Locally bounded linear mappings

and hence
00 00
kykn e|−k βj
{z } ≤ kykn ≤ C kykm e
(kk0 −k )βj km
= C kykm e−km βj .
≤1

In any case we have

00
kykn e−k βj
≤ C max0 kykm e−km βj ,
m≤n

i.e. condition 4.145.2 is satisfied.

( 1 ⇒ 2 ) Let ρ ∈ RN
+ with ρ % +∞ and let
X
kξkk := |ξj |eσk βj with σk := ρ2k
j∈N

the basis of seminorms on λ1∞ (β). By 4.145.2 for k := id we have

∃k 0 ∀n ∃n0 ∃C ≥ 1 ∀y ∈ F ∀j : kykn e−σk0 βj ≤ C max0 kykm e−σm βj .
m≤n

We pick a j0 , such that for j ≥ j0 we have

1 > Ce(σk0 −σk0 +1 )βj (& 0 for j → ∞).
Thus
kykn e−σk0 βj ≤ C max kykm e−σm βj : m ∈ [0, k 0 ] ∪ [n + 1, n0 ] ,


since for k 0 < m ≤ n we have

C kykm e−σm βj ≤ C kykn e−σk0 +1 βj < kykn e−σk0 βj .
Let y ∈ F . Then either
kykn ≤ Ceσk0 βj0 kykk0
or there exists a j ≥ j0 with
kykn e−σk0 βj+1 ≤ Ckykk0 < kykn e−σk0 βj (& 0 for j → ∞)
and, since then for m ≤ k 0
C kykm e−σm βj ≤ C kykk0 < kykn e−σk0 βj ,
the maximum is attained for some m with k 0 < n < m ≤ n0 , i.e.
σm −σ 0
k
 kyk 0  σmb σ−σk0
(σk0 −σm )βj −σk0 βj+1 k k0
kykn ≤ C kykm e ≤ C kykm e bσ 0
k ≤ C kykm C ,
kykn
βj+1
where b := supj βj < ∞. Thus
σ 0 −σ 0
σm − σk 0 1+ nb σ 0 k
kykn1+dm ≤ Cn0 0 kykdkm
0 kykm with dm := and Cn0 0 := C k .
b σk0
Hence in both cases
σ 0 −σ 0
n k
n o
kykn1+dm ≤ Cn000 kykdkm 00
0 kykm , where Cn0 := max (C e
σk0 βj0
) b σk0 , Cn0 0 .

For n ∈ N choose n̄ ≥ n such that dm :=

ρ2m −σk0
b σk 0 ≥ ρm for all m ≥ n̄. By what we
have just shown
∃n̄0 ∃Cn̄000 > 0 ∀y ∃m ∈ [n̄, n̄0 ] : kyk1+ρ
n
m
≤ kyk1+ρ
n̄
m
≤ Cn̄000 kykρkm
0 kykm ,

i.e. 2 is satisfied (with n0 := n̄0 and C 0 := Cn̄000 ).

4.150 Theorem (See [Vog83, Satz 5.2 p.193]).

Let α = (αj )j∈N be a shift-stable sequence and E a Fréchet space with decreasing
0-nbhd basis {Uk : k ∈ N} and Minkowski-functionals k k−k of the polars Uko . Then
1. L(E, λ∞ ∞
∞ (α)) = LB(E, λ∞ (α));

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Locally bounded linear mappings 4.150

⇔ 2. E has the property (LB ∞ ), i.e.

∀ρ ∈ RN 0 0 ∗ 0 ∗ 1+ρm
+ , ρ % ∞ ∀p ∃p ∀n ∃n ∃C ∀x ∃m ∈ [n, n ] : kx k−p0 ≤ C kx∗ kρ−p
m
kx∗ k−m .

( 1 ⇐ 2 ) is valid without the assumption on α and more generally for λ∞

∞ (α) re-
placed by λ∞
∞ (a) with arbitrary a ∈ RJ
≥1 .

Similary as in 4.132 we may assume that p ≤ p0 ≤ n.

Obviously (use 4.137.1 and 4.132.2 ) one has: (Ω) ⇒ (LB∞ ) ⇒ (Ω).
In fact, recall:
(Ω) ∀p ∃p0 ∀n ∀d > 0 ∃C > 0 : k k1+d d
−p0 ≤ C k k−p k k−n .

(Ω) ∀p ∃p0 ∀n ∃C > 0 ∃d > 0 : k k1+d d

−p0 ≤ C k k−p k k−n .

Proof.
( 1 ⇐ 2 ) We will verify condition 4.147.2 :

∀k ∈ NN ∃k 0 ∀n ∃n0 ∃C > 0 ∀j ∀x∗ : aj kx∗ k−k0 ≤ C max0 aj

(n) (m)
kx∗ k−km ,
m≤n
(n)
where aj := enαj . So let w.l.o.g. k % ∞ be given. The property (LB ∞ ) does not
depend on the specific basis of seminorms of E so we may assume that it holds for
the seminorms ||| |||n := k kkn , i.e.
∀ρ ∈ RN 0 0 ∗ 0 ∗ 1+ρ ρ
+ ∀p ∃p ∀n̄ ∃n̄ ∃C ≥ 1 ∀x ∃m ∈ [n̄, n̄ ] : |||x |||−p0
m
≤ C |||x∗ |||−p
m
|||x∗ |||−m .
Now we choose ρ % ∞ such that limm→∞ ρmm = 0 and take p := 0 and obtain a
corresponding p0 . To given n ∈ N we next choose n̄ > n such that n ρm ≤ m − n
for all m ≥ n̄. For each x∗ ∈ E ∗ and j either
enαj |||x∗ |||−p0 ≤ |||x∗ |||−0 ≤ C e0 αj |||x∗ |||−0
or
1+ρ ρ
ρ m
|||x∗ |||−p0 m ≤ C |||x∗ |||−0
m
|||x∗ |||−m < C enαj |||x∗ |||−p0 |||x∗ |||−m
ρ ρ
= C enαj ρm |||x∗ |||−p
m ∗
0 |||x |||−m ≤ C e
(m−n)αj
|||x∗ |||−p
m ∗
0 |||x |||−m

holds. Let k 0 := kp0 , n0 := n̄0 then we have in both cases 4.147.2 :

enαj kx∗ k−k0 = enαj |||x∗ |||−p0 ≤ C max0 emαj |||x∗ |||−m = C max0 emαj kx∗ k−km .
m≤n̄ m≤n

( 1 ⇒ 2 ) Let ρ % ∞ and p ∈ N.
(n)
By 4.147.2 for k : m 7→ p + m and aj := eρn αj we get:
∃p0 > p ∀n ∃n0 ∃C ≥ 1 ∀x∗ ∀j : eρn αj kx∗ k−p0 ≤ C max0 eρm αj kx∗ k−km .
m≤n

For fixed p0 and n > p0 − p choose j0 such that for all j ≥ j0

C < e(ρn −ρn−1 )αj % ∞ for j → ∞

holds. Then
n o
eρn αj kx∗ k−p0 ≤ C max eρm αj kx∗ k−km : m ∈ [0, p0 − p] ∪ [n, n0 ] ,

since for p0 − p < m ≤ n − 1

eρn αj kx∗ k−p0 > C eρn−1 αj kx∗ k−p0 ≥ C eρm αj kx∗ k−(p+m) = C eρm αj kx∗ k−km .
For x∗ ∈ E ∗ , either
C kx∗ k−k0 = C kx∗ k−p ≤ e(ρn −ρp0 −p )αj0 kx∗ k−p0 ,

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4.151 Locally bounded linear mappings

and, since then for m ≤ p0 − p

C eρm αj0 kx∗ k−km ≤ C eρp0 −p αj0 kx∗ k−k0 < eρn αj0 kx∗ k−p0 ,
we get
∃m ∈ [n, n0 ] : eρn αj0 kx∗ k−p0 ≤ C eρm αj0 kx∗ k−km ,
and hence for any d > 0
kx∗ k1+d (ρm −ρn )αj0
kx∗ k−km kx∗ kd−p0 ≤ C e(ρn0 −ρn )αj0 kx∗ k−km kx∗ kd−p .


−p0 ≤ C e

Otherwise, there exists a j ≥ j0 with

e(ρn −ρp0 −p )αj−1 kx∗ k−p0 < Ckx∗ k−p ≤ e(ρn −ρp0 −p )αj kx∗ k−p0 % ∞ for j → ∞

and, since for m ≤ p0 = p

C eρm αj kx∗ k−km ≤ C eρp0 −p αj kx∗ k−k0 < eρn αj kx∗ k−p0 ,
we also get
∃m ∈ [n, n0 ] : kx∗ k−p0 ≤ C e(ρm −ρn )αj kx∗ k−km
ρm −ρn
(ρn −ρp0 −p )αj−1 b
p −p kx∗ k
ρn −ρ 0
≤Ce −km
 kx∗ k d
−p
<C C ∗ kx∗ k−km ,
kx k−p0
−ρn αj
where d := b ρnρm
−ρ 0 with b := supj αj−1 , i.e.
p −p

kx∗ k1+d
−p0 ≤ C
1+d
kx∗ k−km kx∗ kd−p ,
For given n we may now choose the n
from above such that n > max{p0 − p, n} and
b
ρn −ρ 0 ≤ 1 and thus d ≤ ρn −ρb 0 ρm ≤ ρm . Hence, in both cases we have for
p −p p −p

C 0 := C max{e(ρn0 −ρn )αj0 , C d } and n0 := n0 + p the condition 2 :

∃p0 ∀n ∃n0 ∃C > 0 ∃ m ∈ [n, n0 ] : k k1+ρ
−p0
m
≤ C k k−m k kρ−p
m
.

4.151 Corollary (See [Vog83, 6.2. Satz p.198]).

Let E and F be Fréchet spaces. If E has property (LB ∞ ) and F property (DN ),
then
L(E, F ) = LB(E, F ).

Proof. By 4.131 there exists an a : M → R≥1 such that F embeds as closed

subspace into λ∞ ∞
∞ (a). By 4.150 for λ∞ (a) we have

L(E, λ∞ ∞
∞ (a)) = LB(E, λ∞ (a)).

Thus, by 4.141 ,
L(E, F ) = LB(E, F )
as well.

An application of these results is:

Proposition [BD98, Thm. 18 p.23], [BD01, Theorem 3 p.497].

Let F be (F). Then
1. F is (DN);
⇔ 2. C ω (U, F ) = Ctω (U, F ) ∀(∃ ∅ 6=)U ⊆ Rn open.

124 andreas.kriegl@univie.ac.at c July 1, 2016

Locally bounded linear mappings 4.155

Proof. For sake of simplicity we consider only the case U = R treated in [BD98,
Thm. 18 p.23]. By 4.142 : 2 ⇔ L(C ω (R)∗ , F ) = LB(C ω (R)∗ , F ).
(⇐) By [BD98, Proposition 5 p.17] there exists a quotient map q : C ω (R)∗  H(D)
ω
(since Cper (R) ∼
= H(D)) thus L(H(D), F ) = LB(H(D), F ).
= H(D) by [BD98,
(⇒) Let T ∈ L(C ω (R)∗ , F ). Since C ω (R)∗ = limn En with En ∼
−→
Proposition 3 p.16] and H(D) = λ0 (id) by 1.15.6 , there exists for every n ∈ N a
S Un ⊆ En with T (Un ) bounded by 4.146 . By 4.90.1 there are δn > 0 such
0-nbhd
that n δn T (Un ) is bounded. SThus T is bounded on the absolutely convex hull U∞
(which is a 0-nbhd in lim) of n δn Un .
−→
Similarly, the following can be shown:

4.152 Proposition [BD01, Theorem 5 p.498] (See [BD98, Theorem 21 p.24]).

Let F be a complete (LB). Then
1. F ∗ is (Ω);
⇔ 2. C ω (U, F ) = Ctω (U, F ) ∀(∃ ∅ 6=)U ⊆ Rn open.
These results have been generalized to

4.153 Proposition [HH03, Theorem B p.286].

Let F be a Fréchet space having property (LB∞ ) then C ω (U, F ) = Ctω (U, F ) for
every open set U in a Fréchet space E.

4.154 Proposition [HH03, Theorem A p.286].

Let F be a Fréchet space.
1. F is (DN);
⇔ 2. C ω (U, F ) = Ctω (U, F ) ∀U ⊆ E open, where E is (F)+(N)+(Ω̃);
⇔ 3. C ω (U, F ) = Ctω (U, F ) ∀U ⊆ E open, where E is (F)+(S)+(Ω̃) and has an
absolute basis.
A Fréchet space E is said to have property (Ω̃) iff
∀p ∃p0 ∃d > 0 ∀k ∃C > 0 : k k1+d d
−p0 ≤ Ck k−k k k−p .

This property has been used in [DMV84, Theorem 9 p.54] to characterize (NF)
spaces in which not every bounded set is uniformly polar. One has the implications:
(Ω) ⇒ (Ω̃) ⇒ (LB∞ ) ⇒ (Ω).

Another application is:

4.155 Proposition [MV86, 2.3 p.150] and [MV86, 3.4 p.157].

Let E be a Fréchet space. If every entire function f : E → C is of uniformly
bounded type (i.e. there is some 0-nbhd, where the function is bounded on each
multiple) then E satisfies (LB∞ ).
A nuclear Fréchet space E has (Ω) iff every holomorphic functions on polycylindri-
cal U ⊆ E (i.e. finite intersection of sets of the form {x : |x∗ (x)| < 1} for x∗ ∈ E ∗ )
is of uniformly bounded type (i.e. is bounded on each q-bounded subset, which has
positive q-distance to the complement, for some seminorm q for which U is open)

andreas.kriegl@univie.ac.at c July 1, 2016 125

4.158 Locally bounded linear mappings

The subspaces and the quotients of s

Note: Quotient and subspaces of s via (N) and Ext1 = 0 ([Vog84, 2.4 p.362] and
[Vog84, 2.3 p.361]) [Vog84, 2.5 p.363] Quotient and subspaces of s [MV92, 31
p.369],
nuclear-(DN) are the subspaces of s [MV92, 31.5 p.372],
nuclear-(Ω) are the quotients of s [MV92, 31.6 p.373],
nuclear-(DN ∩ Ω) are the direct summand of s [MV92, 31.7 p.375]

4.156 Definition. Vector-valued sequence space s.

Let E be an lcs. Then
s(N, F ) := x ∈ F N : {(1 + n)k xk : n ∈ N} is bounded in F for each k .
n o

Supplied with the norms pk (x) := sup{(1 + n)k p(xn ) : n ∈ N} for k ∈ N and
seminorms p of F it is an lcs and Fréchet if F is Fréchet.

4.157 Proposition. Universal linearizer for s.

= s(N, F ) via T 7→ (T (prn ))n∈N .
Let F be an lcs. Then L(s∗ , F ) ∼

Proof. Let T ∈ L(s∗ , F ) and xn := T (prn ). For k ∈ N and seminorms p of F we

have
pk (xn )n∈N := sup (1 + n)k p(xn ) : n ∈ N = sup p T (1 + n)k prn : n ∈ N

n o n 
 o

n o
≤ sup p(T (x∗ )) : x∗ ∈ Uko ,

since for the standard seminorms (given by kxkk := supn (1 + n)k |xn |, see 1.15.4 )
on s = c0 (A) the polar of the corresponding 0-nbhd Uk is by 1.24

Uko := x∗ ∈ s∗ : kx∗ kUko ≤ 1 = y ∈ KN :

n o n X o
|yn |(1 + n)−k ≤ 1 3 (1 + n)k prn .
n

Thus L(s , F ) → s(N, F ), T 7→ (T (prn ))n∈N , is welldefined, linear, and continuous.

∗

It is bijective, since for x = (xn )n∈N ∈ s(N, F ) the only possible inverse image
T ∈ L(s∗ , F ) is given by
 X  X 
x∗ : x 7→ x∗ (x) = x∗ prn (x) en = x∗ (en ) prn (x) 7→
n n
X  X X
∗ ∗ ∗
7→ T (x ) = T x (en ) prn := x (en ) T (prn ) = x∗ (en ) xn .
n n n
∗ ∗
This
P definition for T makes sense, since any x ∈ S is contained in some Uko , i.e.
∗
n |x (en )|(1 + n)
−k
≤ 1 and {(1 + n)k xn : n ∈ N} is bounded.
Moreover, the so defined T is continuous, since
X  X |x∗ (e )|
n
(p ◦ T )(x∗ ) = p x∗ (en ) xn ≤ k
(1 + n)k p(xn )
n n
(1 + n)
X |x∗ (en )|
≤ k
· sup(1 + n)k p(xn ) ≤ kx∗ kUko pk (x).
n
(1 + n) n

This shows at the same time, that the inverse s(N, F ) → L(s∗ , F ), (xn )n∈N 7→ T ,
is continuous as well.

4.158 s(N, s) ∼
= s (See [MV92, 31.1 p.369]).

126 andreas.kriegl@univie.ac.at c July 1, 2016

The subspaces and the quotients of s 4.160

Proof. Any x = (xn )n∈N ∈ sN is in s(N, s) iff ∀k : {(1 + i)k xi : i ∈ N} is bounded

in s, i.e. ∀k ∀l : {(1 + j)l (1 + i)k xi,j : i, j ∈ N} is bounded in K. Take the bijection
N∼ = N × N, n ↔ (i, j) given by the usual diagonal procedure. Then n is smaller
than the number (m+1)(m+2)2 of lattice points in the triangle with vertices (0, 0),
(m, 0), and (0, m), where m := i + j. And on the other hand i, j ≤ n. Thus
(1 + j)l (1 + i)k ≤ (1 + n)k+l
 k
and (1 + n)k ≤ (1+m)(2+m)
2 ≤ (1 + i + j)2k ≤ (1 + i)2k (1 + j)2k .

So the seminorms of s(N, s) and s can be dominated by each other under this
bijection.

4.159 s → s → sN (See [MV92, 31.3 p.370]).

There is a short exact sequence
0 → s ,→ s  sN → 0.

Proof. By 4.78 (see the proof of 4.131 ) we have the short exact sequence
0 → s ,→ s  KN → 0. By 4.99 the dual sequence 0 → K(N) → s∗ → s∗ → 0 is
Q Q∗

topologically exact and by 4.107 the functor L( , s) is left exact. So we obtain

0 ˆ /
/ s⊗s / s⊗s
ˆ
Q⊗s
// KN ⊗s
ˆ /0
4.61 4.61
0 / L(s∗ , s) / / L(s∗ , s) Q∗∗
/ L(K(N) , s)
4.157
0 / s(N, s) / / s(N, s) / L(K, s)N
4.158
0 /s / /s / / sN /0
In order to see that these isomorphic sequences are short exact we use that any
z ∈ KN ⊗ ˆ π s can be represented by 3.40 as z = n λn xn ⊗ yn with λ ∈ `1 , {xn :
P

n ∈ N} bounded in KN and {yn : n ∈ N} bounded in s. SincePKN is (FM) we find a

set {x̃n : n ∈ N} bounded in s with Q(x̃n ) = xn . Then z̃ := n λn x̃n ⊗ yn ∈ s⊗ ˆ πs
ˆ
with (Q⊗s)(z̃) = z. Since all these tensor products are Fréchet, the top row is a
topologically exact sequence and hence also the bottom row.

4.160 Characterizing the subspaces of s (See [MV92, 31.5 p.372]).

∃ι : E ,→ s ⇔ E is (N)+(F)+(DN).

Proof. (⇒) By 1.15.4 s ∼ = λ∞ (α) with α(n) := ln(n + 1), by 3.78.1 and 4.125
λ∞ (α) is (N) and (DN), and by 3.73.2 and 3.14 E is (N) and (DN).
(⇐) By 4.159 there is an exact sequence 0 → s → s → sN → 0 and by 3.81
there is an embedding E ,→ sN . So the pullback gives another short exact sequence
(where α(n) := ln(n + 1))
 /s / / sN
s O O
1.15.4
? u 3 S ?
λ∞
∞ (α)
/ / s ×sN E //E

which splits by 4.127 . Thus E ,→ s ×sN E ,→ s.

andreas.kriegl@univie.ac.at c July 1, 2016 127

4.163 The subspaces and the quotients of s

4.161 Characterizing the quotients of s (See [MV92, 31.6 p.373]).

∃π : s  E ⇔ E is (N)+(F)+(Ω).

Proof.
(⇒) s ∼
= λ∞ (α) has (N) and (Ω) by 3.78.1 and 4.133.4 . Thus E has (N) and
(Ω) by 3.73.4 and 4.133.2 .
(⇐) By 3.81 there is an embedding E ,→ sN .
Then Q := sN /E is (NF), and thus there exists a 0O 0O
short exact sequence 0 → s −j2→ Q̃ −p2→ Q → 0
0 /E  / sN //Q /0
as in the proof of 4.160 with Q̃ ,→ s and hence O O
Q̃ has (DN) by 3.14.2 . 0 /E /H / Q̃ /0
O O
Let H := {(x, y) ∈ sN × Q̃ : p1 (x) = p2 (x)} be the
pullback. Then the diagram on the right side has sO sO
exact rows and columns and by 4.138 H ∼ = E × Q̃
since E is (Ω). 0 0

Take the left column as top row and proceed anal- 0O 0O

ogously with the sequence from 4.159 as right
0 /s /H / sN /0
column to obtain another diagram with exact rows O O
and columns. Again by 4.138 (or by 4.127 ) /s /G /s /0
G∼ 0 O O
= s × s.
Thus we have quotient mappings sO sO
∼s×s=
s= ∼GH= ∼ E × Q̃  E.
0 0

4.162 Characterizing the complemented subspaces of s (See [MV92, 31.7

p.375]).
⊕
∃ι : E ,→ s ⇔ E (N)+(F)+(DN)+(Ω).
⊕
Here ,→ denotes an embedding as direct summand (i.e. having a left inverse).
Proof.
(⇒) follows from 4.160 and 4.161 .
∼ E × Q̃ ,→ s × s ∼
(⇐) Proceed as in the proof of 4.161 , where H = = s, hence is
(DN). By 4.138 not only the bottom row but also the left column in the second
⊕
diagram split, i.e. s ∼
=s×s∼
=G∼ =H ×s∼ = E × Q̃ × s. Hence E ,→ s.
⊕ ⊕
4.163 s ,→ E ,→ s ⇒ E ∼
= s (See [MV92, 31.2 p.370]).

Proof.
∃E0 : E ∼
= E0 × s and ∃E1 : s ∼
= E × E1 ⇒
⇒ s = E × E1 = E0 × s × E1 ∼
∼ ∼ = E0 × E2 with E2 := s × E1

= s(N, s) ∼
= s(N, E0 ) × s(N, E2 )
4.158
=====
= ⇒s∼
s(N, E0 ) =
∼ E0 × s(N, E0 ) ⇒
= s(N, E0 ) × s(N, E2 ) ∼
⇒s∼ = E0 × s(N, E0 ) × s(N, E2 ) ∼
= E0 × s ∼
=E

128 andreas.kriegl@univie.ac.at c July 1, 2016

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 Index

Ao . . . polar of A, 11 c0 (A), 4
E⊗ ˆ π F . . . completed projective tensor prod- p-approximable operators, 40
uct, 34 p-nuclear operators, 40
E ⊗ F . . . algebraic tensor product, 33 p-summing operators, 40
E ⊗π F . . . projective tensor product, 33 s. . . space of fast falling sequences, 6
E ⊗ε F . . . injective tensor product, 37 b E. . . cbs given by von Neumann bornology,

EA . . . (quotient) space generated by A ⊆ E, 57

2 t F . . . lcs given by bornivorous absolutely con-

EN . . . nuclearification of E, 73 vex subsets, 57

ES . . . Schwartzification of E, 71 (LB ∞ )-space, 123
Ep ∼ = E/ ker p, 2 (LB∞ )-space, 121
H(C). . . space of entire functions, 6 (Ω)-space, 110
H(D). . . space of holomorphic functions on (Ω̃)-space, 125
the unit disk, 6 (Ω)-space, 113
Lequi . . . space of bounded linear functionals (DF)-space, 65
with topology of uniform convergence (DN)-space, 27
on equicontinuous sets, 80 (F)-space, 2
U ⊗V . . . absolutely convex hull of ⊗(U ×V ), (FM)-space, 29
34 (FS)-space, 32
β ∗ (F, F ∗ ). . . topology of uniform convergence (LB)-space, 18, 74
on bounded sets in Fβ∗ , 74 (LF)-space, 18, 74
1
`c , 36 (M)-space, 28
η(E ∗ , E), 77 (N)-space, 38
γ(E ∗ , E), 28

(NF)-space, 53
ιA : EA E. . . canonical injection, 2

(S)-space, 31
ιA : E EA . . . canonical projection, 2 (algebraically) exact sequence, 84
λq (A). . . Köthe sequence space, 4 (bornological) embedding, 85
λpr (α). . . power series space, 6 (bornological) quotient mapping, 85
λp∞ (a). . . generalized power series space, 106 (df)-space, 65
⊗ˆ ε . . . completed injective tensor product, 39 (infra-)countably-barrelled, 63
E-0-sequence, 71 (inverse) limit of lcs, 1
E-nuclear sequence, 72 (quasi-)ℵ0 -barrelled, 63
LB(E, F ). . . space of locally bounded linear (weakly) compact operator, 24
maps, 116
πU,V . . . seminorms of projective tensor prod- absolute basis, 9
uct, 34 absolutely q-summable sequences, 36
lim. . . limit of lcs, 1 absolutely Cauchy sequences, 35
τN . . . topology of uniform convergence on E- absolutely summable sequences, 35
nuclear sequences, 72 absorbing sequence, 69
τS . . . topology of uniform convergence on E- algebraic tensor product, 33
0-sequences, 71 approximation numbers, 26
τc (E ∗ , E). . . topology of uniform convergence approximation property, 80
on compact subsets, 29
τpc (E ∗ , E). . . topology of uniform convergence Banach-disks, 17
on precompact subsets, 29 barrel, 17
ε-product, 81 barrelled space, 17
ε-tensor product, 37 Beurling type, 4
εU,V . . . seminorm of injective tensor prod- bornivorous, 17
uct, 38 bornivorous barrel, 17
lim. . . projective limit of lcs, 1 bornivorous sequence, 69
←−
c0 -barrelled, 60 bornological space, 17

andreas.kriegl@univie.ac.at c July 1, 2016 131

bornology, 57 projective Fréchet space, 100
bounded linear mappings, 17 projective limit, 1
projective tensor product, 33
canonical resolution of a Fréchet space, 91
canonical resolution of a projective limit, 91 quasi complete, 23
Cauchy-net, 2 quasi-normable space, 69
cbs. . . separated convex bornological space, quasi-tonneliert (german), 17
57 quotient seminorms, 3
closed graph theorem, 3
co-nuclear space, 76 rapidly decreasing functions, 4
colimit, 18 reduced inductive limit, 18
compact operators, 80 reduced projective limit, 1
complete, 2, 23 reflexive, 27
completion of the projective tensor product, regular inductive limit, 19
34 right-derived functors, 102
convex bornological space, 57 scalarly absolutely q-summable sequences, 36
coproduct, 18 scalarly absolutely summable sequences, 35
Denjoy-Carleman functions, 4 Schauder-basis, 9
direct sum, 18 Schwartz space, 31
distinguished, 62 Schwartzification, 71
dominating norm, 27 semi-Montel space, 28
semi-reflexive, 27
finite type power series space, 6 seminorms, 1
Fourier-coefficients, 7 separated convex bonrological space, 57
Fréchet space, 2 sequentially complete, 23
shift-stable sequence, 118
generalized power series spaces of infinite type, Silva, 76
106 space of continuous linear mappings, 34
space of continuous multi-linear mappings,
Hermite functions, 8
34
Hermite polynomials, 7
steps of an inductive limit, 18
hypoelliptic PDO, 99
Strict inductive limits, 18
inductive limit, 18 strong topology, 12
infinite type power series space, 6
Tonne (german), 17
infra-c0 -barrelled, 60
tonnelliert (german), 17
infra-barrelled space, 17
topological basis, 9
infra-tonneliert (german), 17
topologically exactsequence, 84
injective Fréchet space, 100
Tschebyscheff(=Chebyshev) polynomials, 9
injective tensor product, 37
ultrabornological space, 17
Köthe sequence space, 4
unconditionally Cauchy summable sequences,
Kelley-space, 3
35
lcs. . . separated locally convex space, 1 universal linearizer, 113
left exact functor, 99 upper semi-continuous, 4
locally bounded linear map, 116
locally complete, 23
locally convex space, 1
locally-complete, 17

Mackey convergent, 18
Mackey-complete, 23
Minkowski-functional, 1, 2
Montel space, 28

nuclear operator, 47
nuclear space, 38
nuclearification, 73

open mapping theorem, 3

polar set, 11
power series space, 6
precompact, 23
probability measure µ, 45

132 andreas.kriegl@univie.ac.at c July 1, 2016