Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-09-24 (gpt-6-sol)
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.

The annihilator detects the closure of a subspace

Statement

Assume the Axiom of Choice. For every linear subspace M⊆X,

M‾=⋂f∈M⊥ker⁡f.

Facts & Assumptions

Given: The Axiom of Choice, A linear subspace M⊆X.

[A1]

AC is used through geometric Hahn–Banach [F1] in step 1.2.

[F1]

Every point outside M‾ is sent to 1 by some continuous functional vanishing on M (Geometric Hahn--Banach theorem for subspaces).

Proof

technique · direct
1.1given

If z∈M‾ and f∈M⊥, continuity of f and f∣M=0 give f(z)=0. Thus M‾ lies in the intersection.

1.2F1given

If z∉M‾, [F1] gives f∈M⊥ with f(z)=1, so z is absent from the intersection.

2.1step 1.1step 1.2∎

The two inclusions prove the formula.

Depends on

Used by

Dependency tree · two levels

9 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