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

Used by

Injective transpose does not imply surjectivity Counterexample
The dual construction reverses arrows Counterexample
The transpose of an injective map need not have norm-dense range Counterexample
Evaluation functionals and point masses Example
Finite-dimensional duals and matrix transposes Example
Transposes of the right and left shifts Example
A lower bound for the transpose forces a dense image of a ball Lemma
Closed range is equivalent to a quotient estimate Lemma
Elementary kernel and range annihilator identities Lemma
Membership in the transpose range by an operator estimate Lemma
The canonical map is natural Lemma
The transpose is bounded with the same norm Lemma
Transposition reverses composition Lemma

Dependency tree · two levels

6 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

Bühler–Salamon, Functional Analysis, §4.1.1, Definition 4.1, p.172 (standard reference, not scraped)