DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

The transpose of a bounded operator

Definition

Let $K = R$ or $C$ . Let $T : X \to Y$ be bounded and linear between normed spaces. Its transpose, or Banach adjoint, is $T^{*} : Y^{*} ⟶ X^{*}, (T^{*} g) (x) = g (T x) .$ The duals are The dual space X^* of a normed space and its dual norm. Composition is bounded by Composition satisfies |ST|\le|S|,|T|, so this has the displayed codomain. It is linear in $g$ over $K$ . No complex conjugation is inserted; a Hilbert adjoint uses a separate inner-product identification.

Depends on

Dependency tree · 2 levels

Sources

Bühler–Salamon, Functional Analysis, §4.1.1, Definition 4.1, p.172