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

Assume the Axiom of Choice. For every linear subspace $M \subseteq X$ ,

$\overline{M} = ⋂_{f \in M^{⊥}} \ker f .$

Given: The Axiom of Choice, A linear subspace $M \subseteq X$ .

[A1]

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

[F1]

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

technique · direct

1.1given

If $z \in \overline{M}$ and $f \in M^{⊥}$ , continuity of $f$ and $f ∣_{M} = 0$ give $f (z) = 0$ . Thus $\overline{M}$ lies in the intersection.

1.2F1given

If $z \notin \overline{M}$ , [F1] gives $f \in M^{⊥}$ 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