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.

Continuous annihilator of a linear subspace

Definition

Let $X$ be a normed space and let $M \subseteq X$ be a linear subspace. Its continuous annihilator is

$M^{⊥} : = {f \in X^{*} : f (m) = 0 for every m \in M} .$

This is an annihilator in the topological dual $X^{*}$ , not in the unrestricted algebraic dual.

Depends on

Used by

The annihilator detects the closure of a subspace Corollary
Distance to a subspace via annihilating functionals Example
Geometric Hahn--Banach theorem for subspaces Theorem

Dependency tree · two levels

10 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 2.52 (standard reference, not scraped)