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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $X$ .

1.1

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.

givenF2

2.1

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. $□$

step 1.1F1algebra

Depends on

Used by

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)