Review of Hilbert and Banach Spaces

Definition 1 (Vector Space) A vector space over C is a set V equipped with two
operations,

(v, w) ∈ V × V 7→ v + w ∈ V (α, v) ∈ C × V 7→ αv ∈ V

called addition and scalar multiplication, respectively, that obey the following axioms.
Additive Axioms. There is an element 0 ∈ V and, for each x ∈ V there is an element
−x ∈ V such that, for all x, y, z ∈ V,
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) 0 + x = x + 0 = x
(iv) (−x) + x = x + (−x) = 0
Multiplicative Axioms. For every x ∈ V and α, β ∈ C,
(v) 0x = 0
(vi) 1x = x
(vii) (αβ)x = α(βx)
Distributive Axioms. For every x, y ∈ V and α, β ∈ C,
(viii) α(x + y) = αx + αy
(ix) (α + β)x = αx + βx

Definition 2 (Subspace) A subset W of a vector space V is called a linear subspace

of V if it is closed under addition and scalar multiplication. That is, if x + y ∈ W and
αx ∈ W for all x, y ∈ W and all α ∈ C. Then W is itself a vector space over C.

Definition 3 (Inner Product)

(a) A inner product on a vector space V is a function (x, y) ∈ V × V 7→ hx, yi ∈ C that

obeys
(i) (Linearity in the second argument) hx, αyi = α hx, yi, hx, y + zi = hx, yi+hx, zi
(ii) (Conjugate symmetry) hx, yi = hy, xi
(iii) (Positive–definiteness) hx, xi > 0 if x 6= 0
for all x, y, z ∈ V and α ∈ C.

(b) Two vectors x and y are said to be orthogonal with respect to the inner product h · , · i
if hx, yi = 0.

(c) We’ll use the terms “inner product space” or “pre–Hilbert space” to mean a vector
space over C equipped with an inner product.

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 1
Definition 4 (Norm)

(a) A norm on a vector space V is a function x ∈ V 7→ kxk ∈ [0, ∞) that obeys

(i) kxk = 0 if and only if x = 0.
(ii) kαxk = |α|kxk
(iii) kx + yk ≤ kxk + kyk
for all x, y ∈ V and α ∈ C.

(b) A sequence vn n∈IN ⊂ V is said to be Cauchy with respect to the norm k · k if

∀ε > 0 ∃N ∈ IN s.t. m, n > N =⇒ kvn − vm k < ε


(c) A sequence vn n∈IN ⊂ V is said to converge to v in the norm k · k if

∀ε > 0 ∃N ∈ IN s.t. n > N =⇒ kvn − vk < ε

(d) A normed vector space is said to be complete if every Cauchy sequence converges.

(e) A subset D of a normed vector space V is said to be dense in V if D = V, where D is

the closure of D. That is, if every element of V is a limit of a sequence of elements of D.

p
Theorem 5 Let h · , ·i be an inner product on a vector space V and set kxk = hx, xi
for all x ∈ V.

(a) The inner product is sesquilinear. That is,

hx, αy + βzi = α hx, yi + β hx, zi
hαx + βy, zi = α hx, zi + β hy, zi
for all x, y, z ∈ V and α, β ∈ C.(1)

(b) kxk is a norm.

(c) The inner product and associated norm obeys

 
(i) (Cauchy–Schwarz inequality)  hx, yi  ≤ kxk kyk
(ii) (Parallelogram law) kx + yk2 + kx − yk2 = 2kxk2 + 2kyk2
(iii) (Polarization identities)
 1
hx, yi = 12 kx + yk2 − kxk2 − kyk2 + 2i kx + iyk2 − kxk2 − kyk2
 1
= 41 kx + yk2 − kx − yk2 + 4i kx + iyk2 − kx − iyk2
for all x, y ∈ V
(1) Physicists and mathematical physicists generally use the convention that inner products are linear
in the second argument and conjugate linear in the first. Some mathematicians use the convention
that inner products are linear in the first argument and conjugate linear in the second.

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 2
Lemma 6 Let k · k be a norm on a vector space V. There exists an inner product h · , · i
on V such that
hx, xi = kxk2 for all x ∈ V

if and only if k · k obeys the parallelogram law.

Definition 7 (Banach Space)

(a) A Banach space is a complete normed vector space.

(b) Two Banach spaces B1 and B2 are said to be isometric if there exists a map U : B1 → B2
that is
(i) linear (meaning that U (αx+βy) = αU (x)+βU (y) for all x, y ∈ B1 and α, β ∈ C)
(ii) onto (a.k.a. surjective)
(iii) isometric (meaning that kU xkB2 = kxkB1 for all x ∈ B1 ). This implies that U is
1–1 (a.k.a. injective).

Definition 8 (Hilbert Space)

(a) A Hilbert space H is a complex inner product space that is complete under the asso-
ciated norm.

(b) Two Hilbert spaces H1 and H2 are said to be isomorphic (denoted H1 ∼

= H2 ) if there
exists a map U : H1 → H2 that is
(i) linear
(ii) onto
(iii) inner product preserving (meaning that hU x, U yiH2 = hx, yiH1 for all x, y ∈ H1 )
Such a map is called unitary.

Theorem 9 (Completion) If V, h · , · iV is any inner product space, then there exists


a Hilbert space H, h ·, , · iH and a map U : V → H such that
(i) U is 1–1
(ii) U is linear
(iii) hU x, U yiH = hx, yiV for all x, y ∈ V
 
(iv) U (V) = U x  x ∈ V is dense in H. If V is complete, then U (V) = H.
H is called the completion of V.

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 3
Example 10
 
(a) Cn = x = (x1 , · · · xn )  x1 , · · · xn ∈ C together with the inner product hx, yi =
Pn
x̄ℓ yℓ is a Hilbert space.
ℓ=1
  P∞
(b) If 1 ≤ p < ∞, then ℓp = (xn )n∈IN  {xn }n∈IN ⊂ C, p
n=1 |xn | < ∞ together with
hP i1/p
∞ p
the norm (xn )n∈IN p = n=1 |xn | is a Banach space.
  P∞
(c) ℓ2 = (xn )n∈IN  |x n | 2
< ∞ is a Hilbert space with the inner product

 P∞n=1
(xn )n∈IN , (yn )n∈IN = n=1 xn yn .
   
(d) ℓ∞ = (xn )n∈IN  sup|xn | < ∞ and c0 = (xn )n∈IN  lim xn = 0 are both Banach
n→∞
n
spaces with the norm (xn )n∈IN ∞ = sup|xn |.

n

(e) Let X be a metric space (or more generally a topological space) and
 
C(X ) = f : X → C  f continuous, bounded
 
C0 (X ) = f : X → C  f continuous, compact support
If X is a subset of IRn or Cn for some n ∈ IN, let
 
C∞ (X ) = f : X → C  f continuous, lim f (x) = 0
|x|→∞

Then C(X ) and C∞ (X ) are Banach spaces with the norm kf k = sup |f (x)|. C0 (X ) is a
x∈X
normed vector space, but need not be complete.

(f) Let 1 ≤ p ≤ ∞. Let (X, M, µ) be a measure space, with X a set, M a σ–algebra and
µ a measure. For p < ∞, set
  R
Lp (X, M, µ) = ϕ : X → C  ϕ is M–measurable and |ϕ(x)|p dµ(x) < ∞
hZ i1/p
p
kϕkp = |ϕ(x)| dµ(x)

For p = ∞, set(2)
 
L∞ (X, M, µ) = ϕ : X → C  ϕ is M–measurable and ess sup |ϕ(x)| < ∞
kϕk∞ = ess sup |ϕ(x)|
This is not quite a Banach space because any function ϕ that is zero almost everywhere
has “norm” zero. So we define an equivalence relation on Lp (X, M, µ) by

ϕ ∼ ψ ⇐⇒ ϕ = ψ a.e.
(2) The essential supremum of |ϕ|, with respect to the measure µ, is denoted ess supx∈X |ϕ(x)| and is
defined by inf{ a ≥ 0 | |ϕ(x)| ≤ a almost everywhere, with respect to µ }.

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 4
As usual, the equivalence class of ϕ ∈ Lp (X, M, µ) is
 
[ϕ] = ψ ∈ Lp (X, M, µ)  ψ ∼ ϕ

Then
 
Lp (X, M, µ) = [ϕ]  ϕ ∈ Lp (X, M, µ)

is a Banach space with

[ϕ] + [ψ] = [ϕ + ψ] a[ϕ] = [aϕ] [ϕ] = kϕkp
p

for all ϕ, ψ ∈ Lp (X, M, µ) and a ∈ C, and L2 (X, M, µ) is a Hilbert space with inner
product Z
h[ϕ], [ψ]i = ϕ(x) ψ(x) dµ(x)

for all ϕ, ψ ∈ L2 (X, M, µ). It is standard to write ϕ in place of [ϕ].

(g) Let D be an open subset of C. Then

  R
A2 (D) = ϕ : D → C  ϕ analytic, D |ϕ(x + iy)|2 dx dy < ∞

is a Hilbert space with the inner product

Z
hϕ, ψi = ϕ(x + iy) ψ(x + iy) dxdy
D

(h) Let ℓ ≥ 0 be an integer and Ω be an open subset of IRn for some n ∈ IN. If α =
(α1 , · · · , αn ) ∈ INn0 , where IN0 = {0} ∪ IN, we shall use ∂ α ϕ(x) to denote the partial
α1 αn
derivative ∂∂xα1 · · · ∂∂xαn ϕ(x). The order of this partial derivative is |α| = α1 + · · · + αn .
1 n
Define
 X Z 
 α 2 n 1/2
kϕkℓ,Ω = ∂ ϕ(x) d x
|α|≤ℓ Ω

for each ϕ ∈ C ℓ (Ω) for which the right hand side is finite. The Sobolev space H ℓ (Ω) is
 
the completion of the vector space ϕ ∈ C ℓ (Ω)  kϕkℓ,Ω < ∞ equipped with the inner
product
X Z
hϕ, ψiℓ,Ω = ∂ α ϕ(x) ∂ α ψ(x) dn x
|α|≤ℓ Ω

Similarly, H0ℓ (Ω) is the completion of C0∞ (Ω).

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 5
Theorem 11 Let −∞ < a < b < ∞ and 1 ≤ p < ∞. The following sets of functions are


dense in Lp [a, b] .
Pn
(a) simple functions (functions of the form j=1 aj χEj (x) with n ∈ IN and the sets Ej
measurable)
Pn
(b) step functions (functions of the form j=1 aj χEj (x) with n ∈ IN and the sets Ej
intervals)
(c) continuous functions that vanish at a and b
(d) periodic C ∞ functions of period b − a
(e) C ∞ functions that are supported in (a, b)

Definition 12 (Basis) Let B be a Banach space and H a Hilbert space.

(a) A subset S of H is an orthonormal subset if each vector in S is of length one and each
pair of distinct vectors in S is orthogonal.

(b) An orthonormal basis (or complete orthonormal system) for H is an orthonormal

subset of H, which is maximal in the sense that it is not properly contained in any other
orthonormal subset of H.

(c) A Schauder basis for B is a sequence en n∈IN of elements of B such that for each
 P∞
v ∈ B there is a unique sequence αn n∈IN ⊂ C such that v = n=1 αn en .

(d) An algebraic basis (or Hamel basis) for B is a subset S ⊂ B such that each x ∈ B has
a unique representation as a finite linear combination of elements of S. This is the case
if and only if every finite subset of S is linearly independent and each x ∈ B has some
representation as a finite linear combination of elements of S.

Theorem 13 Every Hilbert space has an orthonormal basis.

Theorem 14 Every vector space has an algebraic basis.


Theorem 15 Let ei i∈I be an orthonormal basis for the Hilbert space H. Then, for
 
6 0 is countable(3) and
each x ∈ H, i ∈ I  hei , xi =
X X
x= hei , xi ei kxk2 = | hei , xi |2
i∈I i∈I


(The right hand sides converge independent of order.) Conversely, if ci i∈I ⊂ C and
P 2
P
i∈I |ci | < ∞, then i∈I ci ei converges to an element of H.

(3) We’ll include finite in countable.

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 6
Example 16 For each n ∈ ZZ, set

en (x) = √1 einx
2π


Then en n∈ZZ is an orthonormal basis for L2 [0, 2π] .

Definition 17 (Separable) A metric space is said to be separable if it has a countable

dense subset.


Lemma 18 A metric space M, d fails to be separable if and only if there is an ε > 0

and an uncountable subset mi i∈I ⊂ with d(mi , mj ) ≥ ε for all i, j ∈ I with i 6= j.

Theorem 19 Let H be a Hilbert space.

(a) H is separable if and only if it has a countable orthonormal basis.

(b) If dim H = n ∈ IN, then H ∼

= Cn .

(c) If H is separable but is not of finite dimension, then H ∼

= ℓ2 .

Example 20

(a) As L2 ([0, 2π]) has a countable, orthonormal basis, it is separable and isomorphic to ℓ2 .
(S)

(b) ℓ∞ is not separable. To see this define, for each subset S ⊂ IN, x(S) = xn n∈IN ∈ ℓ∞
by n
1 if n ∈ S
xn(S) =
0 if n ∈
/S
This is an uncountable family of elements of ℓ∞ with kx(S) − x(T ) k∞ = 1 for all distinct
subsets S, T of IN.

Definition 21 (Orthogonal Complement) The orthogonal complement, M⊥ , of any

subset M of a Hilbert space H, is defined to be
 
M⊥ = y ∈ H  hy, xi = 0 for all x ∈ M

Theorem 22 Let M be a linear subspace of a Hilbert space H. Then

(a) M⊥ is a closed linear subspace of H.

(b) M ∩ M⊥ = {0}

⊥
(c) M⊥ = M (the closure of M)

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 7
Theorem 23 (Projection) Let M be a closed linear subspace of a Hilbert space H.
Then each x ∈ H has a unique representation x = xk + x⊥ with xk ∈ M and x⊥ ∈ M⊥ .

Definition 24 (Linear Operator) Let B, B′ be Banach spaces and H, H′ be Hilbert

spaces.

(a) Let D be a linear subspace of B. A map A : D → B′ is called a linear operator if it

obeys
A(αx + βy) = αA(x) + βA(y) for all α, β ∈ C and x, y ∈ D

One usually denotes the image of x under A as Ax, rather than A(x). The set D is called
the domain of A and is generally denoted D(A). One often calls A a “linear operator on
B” even when its domain is a proper subset of B.

(b) A linear operator A : D → B′ is said to be bounded if

kAxkB′
kAk = sup = sup kAxkB′ (1)
06=x∈D kxkB x∈D
kxkB =1

is finite. The set of all bounded, linear operators defined on B and taking values in B′ is
denoted L(B, B′ ). With the norm (1), it is itself a Banach space. The set of all bounded,
linear operators defined on B and taking values in B is denoted L(B).

(c) A linear functional on B is a linear operator f : B → C. A bounded linear functional

on B is a linear operator f : B → C for which

|f (x)|
sup
06=x∈B kxkB

is finite.

(d) The dual space of a Banach space B is the space B∗ of all bounded linear functionals
on B. The dual space is itself a Banach space.

(e) Let T : D(T ) ⊂ H → H′ be a linear operator. Denote

 
D(T ∗ ) = ϕ ∈ H′  ∃! η ∈ H s.t. hT ψ, ϕiH′ = hψ, ηiH ∀ ψ ∈ D(T )

If ϕ ∈ D(T ∗ ) the corresponding η is denoted T ∗ ϕ. Thus T ∗ ϕ is the unique vector in H

such that
hT ψ, ϕiH′ = hψ, T ∗ ϕiH for all ψ ∈ D(T )

The operator T ∗ is called the adjoint of T .

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 8
Proposition 25 The normed vector space L(B, B′ ), with the norm (1), is a Banach space.

Lemma 26 Let H be an infinite dimensional Hilbert space. Then there is a linear operator
W : H → H which is defined on all of H, but is not bounded.

Example 27
 
(a) Matrices: Let n ∈ IN. An n × n matrix Mi,j 1≤i,j≤n is naturally associated to the
operator M : Cn → Cn determined by
n
X
(M x)i = Mi,j xj
j=1
 ∗ 
The adjoint operator is the operator so associated to the matrix Mi,j = Mj,i 1≤i,j≤n .

(b) Multiplication Operators: Let 1 ≤ p < ∞, let (X, M, µ) be a measure space and let
f : X → C be measurable. If the essential supremum of f is finite, then

Mf : Lp (X, M, µ) → Lp (X, M, µ)
ϕ(x) 7→ (f ϕ)(x) = f (x) ϕ(x)

is a bounded linear operator with kMf k = ess sup |f (x)|. On the other hand, if the essential
x∈X
supremum of f is infinite, then Mf will not be defined on all of Lp (X, M, µ) and will not
be bounded (as a map into Lp (X, M, µ)). In the case p = 2, Mf∗ = Mf .

(c) Projection Operators: Let H be Hilbert space and let M be a nonempty, closed, linear
subspace of H. Define the map P : H → H by

P x = xk where x = x⊥ + xk is the decomposition of Theorem 23.a

It is a bounded linear operator with kP k = 1, called the orthogonal projection on M. It

obeys
P2 = P P∗ = P
where P ∗ is the adjoint of P . Conversely if P : H → H is a bounded linear operator that
obeys P 2 = P and P ∗ = P , then P is orthogonal projection on M = range(P ).

(d) Integral Operators: Let (X, M, µ) and (Y, N , ν) be measure spaces and T : X ×
Y → C be a function that is measurable with respect to M ⊗ N . Let 1 ≤ p ≤ ∞ and
ϕ ∈ Lp (Y, N , ν). Define, for each x ∈ X for which the function y 7→ T (x, y)ϕ(y) is in
L1 (Y, N , ν), Z
(T ϕ)(x) = T (x, y)ϕ(y) dν(y) (2)
Y

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 9
 (i) If Z
M1 = ess sup |T (x, y)| dν(y) < ∞
x∈X Y
Z
M2 = ess sup |T (x, y)| dµ(x) < ∞
y∈Y X

then (2) defines a bounded operator T : Lp (Y, N , ν) → Lp (X, M, µ) with norm

1− 1 1
kT k ≤ M1 p M2p .
(ii) If the Hilbert–Schmidt norm
hZ i1/2
kT kH.S. = |T (x, y)|2 dµ × ν(x, y)
X×Y

is finite, then (2) defines a bounded operator T : L2 (Y, N , ν) → L2 (X, M, µ) with

norm kT k ≤ kT kH.S. .
In the case p = 2, Z
(T ∗ ψ)(y) = T (x, y)ψ(x) dµ(x)
x

(e) Differential Operators: Let Ω be an open subset of IRn for some n ∈ IN. Recall
that if α = (α1 , · · · , αn ) ∈ INn0 , where IN0 = {0} ∪ IN, we use ∂ α u(x) to denote the
α1 αn
partial derivative ∂∂xα1 · · · ∂∂xαn u(x) and |α| = α1 + · · · + αn to denote the order of this
1 n

partial derivative. For any finite subset I ⊂ INn0 and any family aα (x) α∈I of bounded,
measurable functions on Ω the map
X
ϕ(x) 7→ aα (x) ∂ α ϕ(x)
α∈I

is a linear map on C ∞ (Ω) ⊂ L2 (Ω). But it is not bounded as a map from L2 (Ω) to L2 (Ω).

Lemma 28 Let H be a Hilbert space. Let P : H → H be a bounded operator that obeys

P2 = P P∗ = P

Then P is orthogonal projection on the range of P .

Lemma 29 Let E and F be orthogonal projections onto closed subspaces of a Hilbert

space H. Then E + F is again an orthogonal projection if and only if EF = F E = 0.

Theorem 30 Let V and V ′ be two normed vector spaces and let T : V → V ′ be a linear
transformation. The following are equivalent.
(i) T is continuous at every x ∈ V.
(ii) T is continuous at one x0 ∈ V.
(iii) T is bounded.

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 10
Theorem 31 Let B be a Banach space.

(a) Let S be a subspace of B and λ ∈ S ∗ . Then there is a Λ ∈ B∗ such that kΛkB∗ = kλkS ∗
and Λ(x) = λ(x) for all x ∈ S.
 
(b) Let x ∈ B. There is a nonzero Λ ∈ B∗ such that Λ(x) = kΛkB∗ kxkB .

(c) Let Y be a subspace of B and x ∈ B with the distance from x to Y being d. There is a
Λ ∈ B∗ such that kΛkB∗ ≤ 1, Λ(x) = d and Λ(y) = 0 for all y ∈ Y.

(d) Let x ∈ B. Then

 
kxkB = sup Λ(x)
Λ∈B∗
kΛkB∗ =1

Theorem 32 (The B.L.T. Theorem) Let V be a dense linear subspace of a Banach

space B. Let B′ be a second Banach space and T : V → B′ be a bounded linear transforma-
tion. Then there is a unique bounded linear transformation T̃ : B → B′ such that T x = T̃ x
for all x ∈ V. Furthermore kT k = kT̃ k.

Example 33 We define the Fourier transform as a unitary operator F : L2 (IR) → L2 (IR).

To start we define Schwartz space to be
 
S(IR) = ϕ : IR → C  ϕ is C ∞ , kϕkn,m < ∞ for all integers n, m ≥ 0
 m 
where kϕkn,m = sup xn ddxmϕ (x)
x∈IR

Next we define the Fourier transform and inverse Fourier transform on S(IR) by
Z ∞
ϕ̂(k) = e−ikx ϕ(x) dx
−∞
Z ∞
ψ̌(x) = 1
2π eikx ψ(k) dk
−∞

and verify that the linear functions ϕ 7→ ϕ̂ and ψ 7→ ψ̌ each map S(IR) into (in fact onto)
S(IR) and are inverses of each other and obey
Z ∞ Z ∞
1
ϕ(x) ψ(x) dx = 2π ϕ̂(k) ψ̂(k) dk
−∞ −∞

for all ϕ, ψ ∈ S(IR). Then the B.L.T. theorem provides us with the unique bounded
extension of the map ϕ 7→ ϕ̂ to L2 (IR), which we call F . For the details, see the notes
“Distributions”.

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 11
Theorem 34 (Riesz Representation Theorem) Let H be a Hilbert space and λ ∈ H∗
be a bounded linear functional on H. Then there is a unique yλ ∈ H such that
λ(x) = hyλ , xi
for all x ∈ H. Furthermore kλkH∗ = kyλ kH .

Corollary 35 Let B : H × H → C and C ≥ 0 obey

(i) B(x, αy + βz) = αB(x, y) + βB(x, z)
(ii) B(αx + βy, z) = ᾱB(x, z) + β̄B(y, z)
 
(iii) B(x, y) ≤ Ckxk kyk
for all x, y, z ∈ H and α, β ∈ C. Then there is a unique A ∈ L(H) such that B(x, y) =
hx, Ayi for all x, y ∈ H. Furthermore kAk ≤ C.

Corollary 36 Let H and H′ be Hilbert spaces and T : H → H′ be a bounded linear

operator. Then the adjoint T ∗ of T is a bounded linear operator defined on all of H′ .

Definition 37 (Operator Topologies) Let B and B′ be Banach spaces. Let T : B → B′

and, for each n ∈ IN, Tn : B → B′ be bounded linear operators.

(a) The sequence of Tn n∈IN of operators is said to converge uniformly or in norm to T
if
lim kT − Tn k = 0
n→∞

(b) The sequence of Tn n∈IN of operators is said to converge strongly to T if
lim kT x − Tn xkB′ = 0 for each x ∈ B
n→∞


(c) The sequence of Tn n∈IN of operators is said to converge weakly to T if


∗
lim ℓ Tn x = ℓ T x for each x ∈ B and each ℓ ∈ B′
n→∞

In the event that B′ is a Hilbert space, this is equivalent to

lim hy, Tn xiB′ = hy, T xiB′ for each x ∈ B and each y ∈ B′
n→∞



Remark 38 (Operator Topologies) Since ℓ (Tn − T )x  ≤ kℓkB′∗ (Tn − T )x B′ and

(Tn − T )x ′ ≤ kTn − T k kxkB ,
B

norm convergence =⇒ strong convergence =⇒ weak convergence

In general the other implications are false, unless B and B′ are finite dimensional. This is
illustrated by the following

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 12
Example 39 (Operator Topologies) Let B = B′ = ℓ2 .

(a) Let
n places

z }| {

Pn x1 , x2 , x3 · · · = 0, · · · , 0, xn+1 , xn+2 , xn+3 , · · ·

be projection on the orthogonal complement of the first n components. Then for each
fixed x ∈ ℓ2 , limn→∞ Pn x = 0 so that Pn converges strongly to 0 as n → ∞. But, for any
n > m,
m places
z }| {



(Pn − Pm ) x1 , x2 , x3 · · · = 0, · · · , 0, xm+1 , xm+2 , · · · , xn , 0, · · ·

so that there is a vector x ∈ ℓ2 with (Pn − Pm )x = x. Consequently kPn − Pm k = 1, the

sequence is not Cauchy and does not converge in norm.

(b) Let
n places
z }| {



Rn x1 , x2 , x3 · · · = 0, · · · , 0, x1 , x2 , x3 , · · ·

be right shift by n places. For any x, y ∈ ℓ2

   
 hy, Rn xi  =  hPn y, Rn xi  ≤ kPn yk kRn xk = kPn yk kxk n→∞
−→ 0

So Rn converges weakly to zero as n → ∞. On the other hand, kRn xk = kxk for all n ∈ IN
and x ∈ ℓ2 . So the Rn does not converge strongly or in norm. (If Rn did converge either
weakly
strongly or in norm to some R, the fact that Rn −→ 0 would force R = 0.)

Theorem 40 (Adjoints) Let H be a Hilbert space and S, T ∈ L(H).

(a) The map A 7→ A∗ is a conjugate linear isometric isomorphism of L(H) onto L(H). In
particular
(αA + βB)∗ = αA∗ + βB ∗ kA∗ k = kAk

for all A, B ∈ L(H) and all α, β ∈ C.

(b) (T S)∗ = S ∗ T ∗
∗
(c) (T ∗ ) = T
−1 ∗
(d) If T has a bounded inverse, then T ∗ has a bounded inverse and (T ∗ ) = (T −1 ) .

(e) The map A 7→ A∗ is continuous in the weak and uniform topologies. That is, if
 
An n∈IN converges to A weakly (in norm), then A∗n n∈IN converges to A∗ weakly (in
norm). The map A 7→ A∗ is continuous in the strong topology if and only if H is finite
dimensional.

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 13
(f ) kT ∗ T k = kT k2
 
(g) If T = T ∗ , then kT k = sup | hT x, xi |  x ∈ H, kxk = 1 .

Proposition 41 Let H be a Hilbert space and T : H → H be a bounded linear operator.

(a) We have
|hy,T xi|
kT k = sup kxk kyk
x,vy∈H
x,y6=0

(b) Assume in addition that T = T ∗ . Then

|hx,T xi|
kT k = sup kxk2
x∈H
x6=0

Example 42 Let H = ℓ2 and define the right and left shift operators by

L(x1 , x2 , x3 , · · ·) = (x2 , x3 , · · ·)
R(x1 , x2 , x3 , · · ·) = (0, x1 , x2 , x3 , · · ·)

First observe that kLk = kRk = 1 and that

∞
X ∞
X ∞
X ∞
X
hy, Lxi = yj (Lx)j = yj xj+1 = yi−1 xi = (Ry)i xi = hRy, xi
j=1 j=1 i=2 i=1

so that L∗ = R and R∗ = L. Next observe that, for each n ∈ IN and x ∈ ℓ2 ,

∞
X
n 2 n→∞
kL xk = |xm |2 −→ 0
m=n+1
X∞
kRn xk2 = |xm |2 = kxk2
m=1

Thus, as n → ∞, Ln converges strongly to zero, but Ln ∗ = Rn does not converge strongly

to anything. On the other hand, Ln ∗ does converge weakly to zero since, for all x, y ∈ ℓ2 ,
     
 hy, Ln ∗ xi  =  hy, Rn xi  =  hLn y, xi  ≤ kLn yk kxk n→∞
−→ 0

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 14
Theorem 43 (Principle of Uniform Boundedness etc.) Unless otherwise stated, X
and Y are Banach spaces and T : X → Y is linear and has domain X .

(a) T is bounded if and only if

   
T −1 y ∈ Y  kykY ≤ 1 = x ∈ X  kT xkY ≤ 1

has nonempty interior. (X , Y need not be complete.)

(b) Principle of Uniform Boundedness: Let F ⊂ L(X , Y).

 
If, for each x ∈ X, kT xk  T ∈ F is bounded,
 
then kT k  T ∈ F is bounded,

(Y need not be complete.)

(c) If B : X × Y → C is bilinear and continuous in each variable separately (i.e. B(x, y)

is continuous in x for each fixed y and vice versa), then B(x, y) is jointly continuous (i.e.
if limn→∞ xn = 0 and limn→∞ yn = 0, then limn→∞ B(xn , yn ) = 0.

(d) Open Mapping Theorem: If T ∈ L(X , Y) is surjective (i.e. onto) and if O is an open
 
subset of X , then T O = T x  x ∈ O is an open subset of Y.

(e) Inverse Mapping Theorem: If T ∈ L(X , Y) is bijective (i.e. 1–1 and onto), then T −1
is bounded.

(f ) Closed Graph Theorem: The graph of T is defined to be

 
Γ(T ) = (x, y) ∈ X × Y  y = T x

Then
T is bounded ⇐⇒ Γ(T ) is closed

In other words, T is bounded if and only if

lim xn = x, lim T xn = y =⇒ y = T x
n→∞ n→∞

(g) Hellinger–Toeplitz Theorem: Let T be an everywhere defined linear operator on H that

obeys hx, T yi = hT x, yi for all x, y ∈ H. Then T is bounded.

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 15
References

[Co] John B. Conway, A Course in Functional Analysis, Springer-Verlag, 1990.

[L] Peter D. Lax, Functional Analysis. Wiley, 2002.
[RS] Michael Reed and Barry Simon, Functional Analysis (Methods of modern mathemat-
ical physics, volume 1), Academic Press, 1980.
[Ru] Walter Rudin, Functional Analysis, McGraw-Hill, 1973.

c Joel Feldman. 2011. All rights reserved. September 23, 2011 Review of Hilbert and Banach Spaces 16