DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 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.

Linear hyperplane

Definition

Let $X$ be a vector space over a field $F$ . A linear hyperplane of $X$ is a linear subspace $H$ such that $X / H$ is finite-dimensional over $F$ and $\dim_{F} (X / H) = 1$ . This is algebraic codimension one; closedness is an additional topological property, not part of the definition.

Depends on

Used by

Closed hyperplanes are kernels of nonzero functionals Theorem

Dependency tree · two levels

19 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, Definition 2.46 (standard reference, not scraped)