Weak and Weak* Topologies - Key Properties

The compactness properties of weak and weak* topologies are fundamentally different from norm topologies, providing powerful tools for existence theorems.

TheoremMazur's Theorem

Let $X$ be a normed space and $C \subset X$ a convex set. Then $C$ is weakly closed if and only if $C$ is norm-closed.

Equivalently, the weak closure and norm closure of a convex set coincide.

This remarkable theorem shows that for convex sets, weak and norm topologies produce the same closed sets. This is crucial because many optimization problems involve minimizing convex functionals over convex sets.

TheoremEberlein-Šmulian Theorem

Let $X$ be a Banach space and $K \subset X$ . The following are equivalent:

$K$ is weakly compact
$K$ is weakly sequentially compact
Every sequence in $K$ has a weakly convergent subsequence with limit in $K$

This theorem is powerful because it allows us to work with sequences instead of nets when dealing with weak compactness.

ExampleCompactness in Reflexive Spaces

In a reflexive Banach space $X$ , the closed unit ball $B = \{x : \|x\| \leq 1\}$ is weakly compact. This follows from the Banach-Alaoglu Theorem applied to the bidual.

Consequence: Every bounded sequence in a reflexive space has a weakly convergent subsequence.

TheoremWeak Lower Semicontinuity

Let $X$ be a normed space. The norm $\|\cdot\| : X \to [0, \infty)$ is weakly lower semicontinuous: $\|x\| \leq \liminf_{n \to \infty} \|x_n\|$ whenever $x_n \rightharpoonup x$ .

More generally, any continuous convex function is weakly lower semicontinuous.

Proof

For any $\phi \in X^*$ with $\|\phi\| \leq 1$ : $|\phi(x)| = \lim_{n \to \infty} |\phi(x_n)| \leq \liminf_{n \to \infty} \|x_n\|$

Taking the supremum over all such $\phi$ gives $\|x\| \leq \liminf \|x_n\|$ .

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ExampleApplication to Minimization

Consider minimizing $J(x) = \|x\|^2$ over a weakly closed convex set $K$ in a reflexive space.

Any minimizing sequence $(x_n)$ with $J(x_n) \to \inf_K J$ is bounded, so has a weakly convergent subsequence $x_{n_k} \rightharpoonup x_0 \in K$ (by Mazur).

By weak lower semicontinuity: $J(x_0) \leq \liminf J(x_{n_k}) = \inf_K J$

Thus $x_0$ is a minimizer.

Remark

The combination of weak compactness and weak lower semicontinuity of convex functionals forms the foundation of the direct method in the calculus of variations, providing existence of minimizers for a vast class of variational problems.

These properties make weak topologies indispensable in modern analysis, particularly in optimization and PDE theory.