DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-09-24 (gpt-6-sol)

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.

Reflexivity is surjectivity of the canonical map

Definition

Let $K = R$ or $C$ . A Banach space $X$ is reflexive if its canonical map $J_{X} : X \to X^{* *}$ from The canonical evaluation map into the bidual is surjective. Assuming the Axiom of Choice (The Axiom of Choice), The canonical bidual map is an isometry shows that this map is already an isometric embedding. Surjectivity means that every bounded linear functional on $X^{*}$ is evaluation at a vector of $X$ . Merely specifying some isomorphism between $X$ and $X^{* *}$ is not this definition.

Depends on

Dependency tree · 2 levels

Sources

Bühler–Salamon, Functional Analysis, Definition 2.70, p.89