LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

An open convex neighbourhood is recovered from its gauge

Statement

If $U \subseteq X$ is open, convex, and $0 \in U$ , then $U$ is absorbing and

$U = {x \in X : p_{U} (x) < 1} .$

Facts & Assumptions

Given: An open convex $U \subseteq X$ containing $0$ .

[F1]

The gauge is defined for absorbing sets by an infimum over positive dilates (Minkowski functional of an absorbing set).

Proof

technique · direct

1.1givenF1choose

Openness gives $B (0, r) \subseteq U$ for some $r > 0$ . For any $x$ , $x \in (∥ x ∥ / r + 1) U$ , so $U$ is absorbing and [F1] applies.

2.1step 1.1F1given

If $x \in U$ , openness gives $ε > 0$ with $(1 + ε) x \in U$ ; hence $p_{U} (x) \leq (1 + ε)^{- 1} < 1$ .

3.1step 1.1F1choose∎

If $p_{U} (x) < 1$ , choose $t < 1$ with $x \in t U$ , say $x = t u$ . Convexity and $0 \in U$ imply $t u \in U$ . Thus both inclusions hold.

Depends on

Used by

Separate a point from an open convex set Theorem

Dependency tree · two levels

16 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, Lemma 5.1 (standard reference, not scraped)