TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-09-23 (gpt-6-sol)

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Mazur theorem: weak and norm closure agree for convex sets

Statement

Assume the Axiom of Choice. For a convex subset $C$ of a normed space $X$ , its norm closure equals its weak closure. Here the weak topology is generated by the basic neighbourhoods of $x$

${y : ∣ f_{j} (y - x) ∣ < ε (1 \leq j \leq n)}$

for finitely many $f_{j} \in X^{*}$ and some $ε > 0$ .

Facts & Assumptions

Given: The Axiom of Choice, a convex set $C \subseteq X$ , and $D = {\overline{C}}^{∥ \cdot ∥}$ .

[F1]

Disjoint convex sets, one open, are strictly separated by a nonzero continuous functional (Separation of disjoint convex sets when one is open).

[A1]

The Axiom of Choice is used through the Hahn–Banach extension behind [F1].

Proof

technique · direct

1.1given

Every $f \in X^{*}$ is norm-continuous, so every displayed basic weak neighbourhood is norm-open. Thus every weak-open set is norm-open, and $D \subseteq {\overline{C}}^{w e a k}$ .

1.2givenF1A1choose

If $C = \emptyset$ , both closures are empty. Otherwise $D$ is nonempty; if $x \notin D$ , choose $r > 0$ with $x \notin D + B (0, r)$ . This latter set is open and convex, so under [A1], [F1] yields $f \neq 0$ with $Re f (d + b) < Re f (x)$ for $d \in D$ , $∥ b ∥ < r$ .

2.1step 1.2algebra

Taking suprema over $b \in B (0, r)$ gives $\sup_{d \in D} Re f (d) + r ∥ f ∥ \leq Re f (x)$ . The weak neighbourhood ${y : ∣ f (y - x) ∣ < r ∥ f ∥ / 2}$ therefore misses $D$ , hence misses $C$ .

3.1step 1.1step 2.1∎

Thus $x$ is not in the weak closure whenever it is not in $D$ , proving the reverse inclusion and equality.

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Topics in Real and Functional Analysis, Theorem 5.11 (standard reference, not scraped)