CorollaryStatement: 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.

A closed convex set is an intersection of closed half-spaces

Statement

Every nonempty closed convex $C \subseteq X$ is the intersection of the closed affine half-spaces that contain $C$ .

Facts & Assumptions

Given: A nonempty closed convex set $C \subseteq X$ .

[F1]

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

Proof

technique · direct

1.1

Since every closed affine half-space in the family contains $C$ , their intersection contains $C$ .

given

1.2

If $x \notin C$ , choose $r > 0$ with $B (x, r) \cap C = \emptyset$ . The set $C + B (0, r / 2)$ is open and convex and still omits $x$ ; [F1] separates it from ${x}$ .

givenF1choose

2.1

The closed half-space $H_{x} = {z : Re f (z) \leq Re f (x) - ε}$ obtained by taking a positive fraction of the separation gap contains $C$ and excludes $x$ . Thus every exterior point is excluded from the intersection, proving equality.

step 1.2algebra∎

Depends on

Used by

Mazur theorem: weak and norm closure agree for convex sets Theorem

Dependency tree · two levels

8 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

Theo Buehler and Dietmar Salamon, Functional Analysis, Exercise 2.51 (standard reference, not scraped)