60 3 Weak Topologies. Reflexive Spaces. Separable Spaces.
Uniform Convexity
In order to complete the proof of (1) it suffices to know that BE is closed in the
topology σ (E, E ⋆ ). But we have
!
BE = {x ∈ E; |⟨f, x⟩| ≤ 1},
f ∈E ⋆
∥f ∥≤1
which is an intersection of weakly closed sets.
Example 2. The unit ball U = {x ∈ E; ∥x∥ < 1}, with E infinite-dimensional, is
never open in the weak topology σ (E, E ⋆ ). Suppose, by contradiction, that U is
weakly open. Then its complement U c = {x ∈ E; ∥x∥ ≥ 1} is weakly closed. It
follows that S = BE ∩ U c is also weakly closed; this contradicts Example 1.
⋆ Remark 3. In infinite-dimensional spaces the weak topology is never metrizable,
i.e., there is no metric (and a fortiori no norm) on E that induces on E the weak
topology σ (E, E ⋆ ); see Exercise 3.8. However, as we shall see later (Theorem 3.29),
if E ⋆ is separable one can define a norm on E that induces on bounded sets of E the
weak topology σ (E, E ⋆ ).
⋆ Remark 4. Usually, in infinite-dimensional spaces, there exist sequences that con-
verge weakly and do not converge strongly. For example, if E ⋆ is separable or if E
is reflexive one can construct a sequence (xn ) in E such that ∥xn ∥ = 1 and xn ⇀ 0
weakly (see Exercise 3.22). However, there are infinite-dimensional spaces with the
property that every weakly convergent sequence is strongly convergent. For exam-
ple, ℓ1 has that unusual property (see Problem 8). Such spaces are quite “rare” and
somewhat “pathological.” This strange fact does not contradict Remark 2, which as-
serts that in infinite-dimensional spaces, the weak topology and the strong topology
are always distinct: the weak topology is strictly coarser than the strong topology.
Keep in mind that two metric (or metrizable) spaces with the same convergent se-
quences have identical topologies; however, if two topological spaces have the same
convergent sequences they need not have identical topologies.
3.3 Weak Topology, Convex Sets, and Linear Operators
Every weakly closed set is strongly closed and the converse is false in infinite-
dimensional spaces (see Remark 2). However, it is very useful to know that for
convex sets, weakly closed = strongly closed:
• Theorem 3.7. Let C be a convex subset of E. Then C is closed in the weak topology
σ (E, E ⋆ ) if and only if it is closed in the strong topology.
Proof. Assume that C is closed in the strong topology and let us prove that C is
closed in the weak topology. We shall check that the complement C c of C is open in
the weak topology. To this end, let x0 ∈
/ C. By Hahn–Banach there exists a closed
3.3 Weak Topology, Convex Sets, and Linear Operators 61
hyperplane strictly separating {x0 } and C. Thus, there exist some f ∈ E ⋆ and some
α ∈ R such that
⟨f, x0 ⟩ < α < ⟨f, y⟩ ∀y ∈ C.
Set
V = {x ∈ E; ⟨f, x⟩ < α};
so that x0 ∈ V , V ∩ C = ∅ (i.e., V ⊂ C c ) and V is open in the weak topology.
Corollary 3.8 (Mazur). Assume (xn ) converges weakly to x. Then there exists a
sequence (yn ) made up of convex combinations of the xn ’s that converges strongly
to x.
Proof. Let C = conv(∪∞ p=1 {xp }) denote the convex hull of the xn ’s. Since x belongs
to the weak closure of ∪∞
p=1 {xp } it belongs a fortiori to the weak closure of C. By
Theorem 3.7, x ∈ C, the strong closure of C, and the conclusion follows.
Remark 5. There are some variants of Corollary 3.8 (see Exercises 3.4 and 5.24).
Also, note that the proof of Theorem 3.7 shows that every closed convex set C
coincides with the intersection of all the closed half-spaces containing C.
• Corollary 3.9. Assume that ϕ : E → (−∞ + ∞] is convex and l.s.c. in the strong
topology. Then ϕ is l.s.c. in the weak topology σ (E, E ⋆ ).
Proof. For every λ ∈ R the set
A = {x ∈ E; ϕ(x) ≤ λ}
is convex and strongly closed. By Theorem 3.7 it is weakly closed and thus ϕ is
weakly l.s.c.
• Remark 6. It may be rather difficult in practice to prove that a function is l.s.c. in
the weak topology. Corollary 3.9 is often used as follows:
ϕ convex and strongly continuous ⇒ ϕ weakly l.s.c.
For example, the function ϕ(x) = ∥x∥ is convex and strongly continuous; thus it is
weakly l.s.c. In particular, if xn ⇀ x weakly, it follows that ∥x∥ ≤ lim inf ∥xn ∥ (see
also Proposition 3.5).
Theorem 3.10. Let E and F be two Banach spaces and let T be a linear operator
from E into F . Assume that T is continuous in the strong topologies. Then T is
continuous from E weak σ (E, E ⋆ ) into F weak σ (F, F ⋆ ) and conversely.
Proof. In view of Proposition 3.2 it suffices to check that for every f ∈ F ⋆ the map
x 0→ ⟨f, T x⟩ is continuous from E weak σ (E, E ⋆ ) into R. But the map x 0→ ⟨f, T x⟩
is a continuous linear functional on E. Therefore, it is also continuous in the weak
topology σ (E, E ⋆ ).
62 3 Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity
Conversely, suppose that T is continuous from E weak into F weak. Then G(T )
is closed in E × F equipped with the product topology σ (E, E ⋆ ) × σ (F, F ⋆ ), which
is clearly the same as σ (E × F, (E × F )⋆ ). It follows that G(T ) is strongly closed
(any weakly closed set is strongly closed). We conclude with the help of the closed
graph theorem (Theorem 2.9) that T is continuous from E strong into F strong.
Remark 7. The argument above shows more: that if a linear operator T is continuous
from E strong into F weak then T is continuous from E strong into F strong. As
a consequence, for linear operators, the following continuity properties are all the
same: S → S, W → W , S → W (S = strong, W = weak). On the other hand,
very few linear operators are continuous W → S; this happens if and only if T is
continuous S → S and, moreover, dim R(T ) < ∞ (see Exercise 6.7).
Also, note that in general, nonlinear maps that are continuous from E strong into
F strong are not continuous from E weak into F weak (see, e.g., Exercise 4.20).
This is a major source of difficulties in nonlinear problems.
3.4 The Weak⋆ Topology σ (E ⋆ , E)
So far, we have two topologies on E ⋆ :
(a) the usual (strong) topology associated to the norm of E ⋆ ,
(b) the weak topology σ (E ⋆ , E ⋆⋆ ), obtained by performing on E ⋆ the construction
of Section 3.3.
We are now going to define a third topology on E ⋆ called the weak⋆ topology and
denoted by σ (E ⋆ , E) (the ⋆ is here to remind us that this topology is defined only on
dual spaces). For every x ∈ E consider the linear functional ϕx : E ⋆ → R defined
by f 0 → ϕx (f ) = ⟨f, x⟩. As x runs through E we obtain a collection (ϕx )x∈E of
maps from E ⋆ into R.
Definition. The weak⋆ topology, σ (E ⋆ , E), is the coarsest topology on E ⋆ associated
to the collection (ϕx )x∈E (in the sense of Section 3.1 with X = E ⋆ , Yi = R, for all
i, and I = E).
Since E ⊂ E ⋆⋆ , it is clear that the topology σ (E ⋆ , E) is coarser than the topology
σ (E ⋆ , E ⋆⋆ ); i.e., the topology σ (E ⋆ , E) has fewer open sets (resp. closed sets) than
the topology σ (E ⋆ , E ⋆⋆ ), which in turn has fewer open sets (resp. closed sets) than
the strong topology.
Remark 8. The reader probably wonders why there is such hysteria over weak topolo-
gies! The reason is the following: a coarser topology has more compact sets. For
example, the closed unit ball BE ⋆ in E ⋆ , which is never compact in the strong topol-
ogy (unless dim E < ∞; see Theorem 6.5), is always compact in the weak⋆ topology
(see Theorem 3.16). Knowing the basic role of compact sets—for example, in exis-
tence mechanisms such as minimization—it is easy to understand the importance of
the weak⋆ topology.
