Weak and Weak* Topologies - Examples and Constructions

Reflexivity provides a deep connection between weak and weak* topologies, characterizing when a Banach space can be identified with its bidual.

TheoremCharacterization of Reflexivity

A Banach space $X$ is reflexive if and only if the closed unit ball is weakly compact.

This provides a topological characterization of reflexivity. The weakly compact spaces are exactly those where bounded sets have good compactness properties.

Proof

Forward: If $X$ is reflexive, then $J : X \to X^{**}$ is surjective. The unit ball $B_X$ maps onto $B_{X^{**}}$ , which is weak* compact by Banach-Alaoglu. Since $J$ is a homeomorphism for the weak and weak* topologies, $B_X$ is weakly compact.

Reverse: If $B_X$ is weakly compact, we show $X^{**} = J(X)$ . For any $x^{**} \in B_{X^{**}}$ , by Goldstine's theorem, there exists a net $(x_\alpha)$ in $B_X$ with $J(x_\alpha) \to x^{**}$ weak*. If $B_X$ is weakly compact, we can extract a subnet converging weakly (hence weak*) to some $x \in B_X$ . Thus $x^{**} = J(x)$ .

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ExampleReflexive Spaces

Hilbert Spaces: All Hilbert spaces are reflexive
$L^p$ Spaces: For $1 < p < \infty$ , $L^p(\mu)$ is reflexive
Sobolev Spaces: $W^{k,p}(\Omega)$ for $1 < p < \infty$ is reflexive
Non-Reflexive: $L^1$ , $L^\infty$ , $C[0,1]$ , $\ell^1$ , $\ell^\infty$ , $c_0$ are not reflexive

TheoremWeak Compactness Criterion

Let $X$ be a Banach space and $K \subset X$ . Then $K$ is relatively weakly compact if and only if:

$K$ is bounded
For every $\varepsilon > 0$ , there exists a finite-dimensional subspace $F \subset X$ such that for every $x \in K$ , $\text{dist}(x, F) < \varepsilon$

DefinitionSchauder Basis

A sequence $(e_n)$ in a Banach space $X$ is a Schauder basis if every $x \in X$ has a unique representation $x = \sum_{n=1}^\infty \alpha_n e_n$ with convergence in norm.

If $X$ has a Schauder basis, then $X$ is separable.

ExampleWeak* vs Weak Convergence

In non-reflexive spaces, weak* convergence is genuinely weaker than weak convergence.

Consider $X = c_0$ with dual $X^* = \ell^1$ . The sequence $(\phi_n)$ in $\ell^1$ defined by $\phi_n = e_n$ (standard basis) converges weak* to $0$ (since $\phi_n(x) = x_n \to 0$ for all $x \in c_0$ ), but does not converge weakly in $\ell^1$ (the dual of $\ell^1$ is $\ell^\infty$ , and there exist bounded sequences where $\sum_n x_n$ doesn't converge).

Remark

Reflexive spaces are particularly well-behaved for analysis because weak compactness, sequential weak compactness, and weak* compactness all coincide. In non-reflexive spaces, these concepts diverge, requiring more careful analysis.

Understanding reflexivity and weak compactness is essential for applying functional analytic methods to variational problems and PDEs.