TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Weak topology is hausdorff

Statement

Assume HB, the real dominated-extension principle of The real dominated-extension principle as an additional hypothesis over ZF. The weak topology of every real or complex normed space $X$ is Hausdorff.

Facts & Assumptions

[F1]

The finite disk sets are weak neighborhoods (Basic weak neighborhoods).

[F2]

Under HB, for each $v \neq 0$ there is $f \in X^{*}$ with $∥ f ∥ = 1$ and $f (v) = ∥ v ∥$ (Relative dual norming, point separation, and recovery of the norm).

Proof

Given: HB and a real or complex normed space $X$ .

1.1givenF2

Fix distinct $x, y \in X$ . Apply dual norming to $v = x - y \neq 0$ to obtain $f \in X^{*}$ with $∣ f (x) - f (y) ∣ = ∥ x - y ∥ = : d > 0$ . This is the only use of HB; no simultaneous selection over pairs is needed.

2.1step 1.1F1algebra

Set $U = {z : ∣ f (z - x) ∣ < d / 3}$ and $V = {z : ∣ f (z - y) ∣ < d / 3}$ . These are weak neighborhoods of $x$ and $y$ . If $z$ belonged to both, the triangle inequality would give $d \leq ∣ f (x - z) ∣ + ∣ f (z - y) ∣ < 2 d / 3$ , impossible. Thus $U \cap V = \emptyset$ . Every distinct pair is separated; if $X = {0}$ there is no such pair. $□$

Depends on

Used by

Bounded Sobolev sequences have strongly convergent subsequences with the weak limit as limit Corollary
Weak closure can exceed sequential weak closure Counterexample

Dependency tree · two levels

11 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

Bühler–Salamon, Functional Analysis (2017); exact harvest in batch coverage (standard reference, not scraped)
Teschl, Topics in Real and Functional Analysis (2017); exact harvest in batch coverage (standard reference, not scraped)