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.

The annihilator detects the closure of a subspace

Statement

For every linear subspace $M \subseteq X$ ,

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

Facts & Assumptions

Given: A linear subspace $M \subseteq X$ .

[F1]

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

Proof

technique · direct

1.1

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.

given

1.2

If $z \notin \overline{M}$ , [F1] gives $f \in M^{⊥}$ with $f (z) = 1$ , so $z$ is absent from the intersection.

F1given

2.1

The two inclusions prove the formula.

step 1.1step 1.2∎

Depends on

Used by

Density is characterized by a zero annihilator Corollary

Dependency tree · two levels

7 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, Corollary 2.55 (standard reference, not scraped)