Alphabeta Math
TheoremStatement: 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.

Closed hyperplanes are kernels of nonzero functionals

Statement

A linear hyperplane H⊆X is closed if and only if H=ker⁡f for some nonzero f∈X∗.

Facts & Assumptions

Given: A linear hyperplane H⊆X.

[F1]

A point outside the closure of a subspace is separated from it by a continuous functional vanishing on that subspace (Geometric Hahn--Banach theorem for subspaces).

Proof

technique · direct
1.1

Suppose H is closed and choose x∉H. By [F1] there is f∈X∗ with f∣H=0 and f(x)=1. Thus H⊆ker⁡f.

givenF1choose
2.1

Since X/H has dimension one, a proper subspace containing H cannot strictly contain H. As f(x)=1, ker⁡f is proper; hence ker⁡f=H.

step 1.1given
3.1

Conversely, if H=ker⁡f with f∈X∗ nonzero, continuity makes H closed. The induced nonzero map X/H→K is injective and onto, so dim⁡(X/H)=1.

givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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