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.

Density is characterized by a zero annihilator

Statement

A linear subspace $M \subseteq X$ is dense if and only if $M^{⊥} = {0}$ .

Facts & Assumptions

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

[F1]

$\overline{M} = ⋂_{f \in M^{⊥}} \ker f$ (The annihilator detects the closure of a subspace).

Proof

technique · direct

1.1

If $M$ is dense, [F1] says every $f \in M^{⊥}$ vanishes on $X$ , hence is zero.

F1given

2.1

If $M^{⊥} = {0}$ , the intersection in [F1] is $X$ , so $\overline{M} = X$ and $M$ is dense.

F1given∎

Depends on

The annihilator detects the closure of a subspace

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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 2.56 (standard reference, not scraped)