Chapter 35

Weak Derivatives35.1 Weak ∗ Convergence

A very important sort of convergence in applications of functional analysis is the conceptof weak or weak ∗ convergence. It is important because it allows you to assert the exis-tence of a convergent subsequence of a given bounded sequence. The only problem is theconvergence is very weak so it does not tell you as much as you would like. Nevertheless,it is a very useful concept. The big theorems in the subject are the Eberlein Smulian theo-rem and the Banach Alaoglu theorem about the weak or weak ∗ compactness of the closedunit balls in either a Banach space or its dual space. These theorems are proved in Yosida[127]. Here I will present a special case which turns out to be by far the most important inapplications and it is not hard to get from the Riesz representation theorem for Lp. First Idefine weak and weak ∗ convergence.

Definition 35.1.1 Let X ′ be the dual of a Banach space X and let {x∗n} be a sequence ofelements of X ′. Then x∗n converges weak ∗ to x∗ if and only if for all x ∈ X,

limn→∞

x∗n (x) = x∗ (x) .

A sequence in X ,{xn} converges weakly to x ∈ X if and only if for all x∗ ∈ X ′

limn→∞

x∗ (xn) = x∗ (x) .

The main result is contained in the following lemma.

Lemma 35.1.2 Let X ′ be the dual of a Banach space, X and suppose X is separable. Thenif {x∗n} is a bounded sequence in X ′, there exists a weak ∗ convergent subsequence.

Proof: Let D be a dense countable set in X . Then the sequence, {x∗n (x)} is boundedfor all x and in particular for all x ∈ D. Use the Cantor diagonal process to obtain a sub-sequence, still denoted by n such that x∗n (d) converges for each d ∈ D. Now let x ∈ X becompletely arbitrary. In fact {x∗n (x)} is a Cauchy sequence. Let ε > 0 be given and pickd ∈ D such that for all n

|x∗n (x)− x∗n (d)|<ε

3.

This is possible because D is dense. By the first part of the proof, there exists Nε such thatfor all m,n > Nε ,

|x∗n (d)− x∗m (d)|< ε

3.

Then for such m,n,

|x∗n (x)− x∗m (x)| ≤ |x∗n (x)− x∗n (d)|+ |x∗n (d)− x∗m (d)|

+ |x∗m (d)− x∗m (x)| <ε

3+

ε

3+

ε

3= ε.

Since ε is arbitrary, this shows {x∗n (x)} is a Cauchy sequence for all x ∈ X .

1231