Hahn-Banach and Dual Spaces - Main Theorem

The Goldstine Theorem provides a fundamental result about the relationship between a Banach space and its bidual, showing that every Banach space is weak* dense in its bidual.

TheoremGoldstine Theorem

Let $X$ be a Banach space with natural embedding $J : X \to X^{**}$ . Then $J(X)$ is weak* dense in $X^{**}$ . More precisely, for any $x^{**} \in X^{**}$ , $\varepsilon > 0$ , and finite set $\{\phi_1, \ldots, \phi_n\} \subset X^*$ , there exists $x \in X$ with $\|x\| \leq \|x^{**}\|$ such that $|x^{**}(\phi_i) - J(x)(\phi_i)| = |\phi_i(x) - x^{**}(\phi_i)| < \varepsilon$ for all $i = 1, \ldots, n$ .

This theorem shows that even for non-reflexive spaces, the original space $X$ is "almost" the bidual when viewed in the weak* topology.

Proof

Fix $x^{**} \in X^{**}$ with $\|x^{**}\| = 1$ , $\varepsilon > 0$ , and $\phi_1, \ldots, \phi_n \in X^*$ . Let $M = \text{span}\{\phi_1, \ldots, \phi_n\} \subset X^*$ .

Define $f : M \to \mathbb{K}$ by $f(\phi) = x^{**}(\phi)$ . Since $x^{**}$ is a bounded linear functional on $X^*$ , we have $|f(\phi)| \leq \|\phi\|$ for $\phi \in M$ .

By Hahn-Banach, extend $f$ to $F \in X^{**}$ with $\|F\| = \|f\| \leq 1$ . Since the weak* closure of the unit ball $B_X = \{x \in X : \|x\| \leq 1\}$ in $X^{**}$ equals $B_{X^{**}}$ (by Banach-Alaoglu), and $F \in B_{X^{**}}$ , there exists a net $(x_\alpha)$ in $B_X$ such that $J(x_\alpha) \to F$ weak*.

For sufficiently large $\alpha$ , we have $|J(x_\alpha)(\phi_i) - F(\phi_i)| < \varepsilon$ for all $i$ . Since $F(\phi_i) = f(\phi_i) = x^{**}(\phi_i)$ , we obtain the result.

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ExampleConsequences

Density: Every Banach space is weak* dense in its bidual
Approximation: Elements of $X^{**}$ can be approximated by elements of $X$ in the weak* topology
Reflexivity Criterion: $X$ is reflexive if and only if the closed unit ball of $X$ is weakly compact

Remark

The Goldstine Theorem is crucial for understanding the structure of non-reflexive spaces. It shows that reflexivity fails not because $X$ is "too small" compared to $X^{**}$ in the weak* topology, but rather because certain weak* limits of sequences in $X$ lie outside $X$ .

This theorem has applications in optimization, where minimizing sequences may converge in $X^{**}$ but not in $X$ , leading to the study of relaxed problems.