CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-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.

Dense range is equivalent to injectivity of the transpose

Statement

Assume the Axiom of Choice. Let $K = R$ or $C$ . For a bounded linear map $T : X \to Y$ between normed spaces, ${\overline{ran T}}^{∥ \cdot ∥} = Y ⟺ \ker T^{*} = {0} .$

Facts & Assumptions

Given: The Axiom of Choice, The spaces, maps, scalar field, and hypotheses in the statement above. All duals consist of linear functionals over the ambient field; evaluation has no conjugation.

[A1]

AC is used through the primal annihilator-closure clause of [F1] in step 2.1.

[F1]

From Elementary kernel and range annihilator identities, with its stated hypotheses: Let $K = R$ or $C$ . For a bounded linear $T : X \to Y$ between normed spaces, $(ran T)^{⊥} = \ker T^{*},^{⊥} (ran T^{*}) = \ker T, {\overline{ran T}}^{∥ \cdot ∥} =^{⊥} (\ker T^{*}) .$ The closure in the last identity is in $Y$ .

Proof

1.1F1

If the range is dense and $g \in \ker T^{*}$ , then $g$ vanishes on the range by the elementary identity. Continuity makes it vanish on $Y$ , so $g = 0$ .

2.1F1∎

If $\ker T^{*} = {0}$ , the third elementary identity gives $\overline{ran T} =^{⊥} {0} = Y$ . This also handles $Y = {0}$ and the zero operator whenever the criterion holds.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, Theorem 4.8(ii), p.174 (standard reference, not scraped)