CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-26 (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.

Quantitative lifting form of the open mapping theorem

Statement

Assume DC. For a surjective bounded linear $T : X \to Y$ between Banach spaces, some $c > 0$ satisfies $c B_{Y} (0, 1) \subseteq T (B_{X} (0, 1))$ .

Facts & Assumptions

Given: DC and $T$ as in the statement.

[A1]

The stated Dependent Choice is the premise for the open-mapping theorem (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

Proof

technique · direct

1.1givenA1

Under [A1], Open mapping theorem makes $T (B_{X} (0, 1))$ an open neighbourhood of $0$ .

2.1step 1.1∎

It therefore contains $B_{Y} (0, c)$ for some $c > 0$ , which is exactly the claim.

Depends on

Used by

Surjectivity is equivalent to a lower bound for the transpose Theorem

Dependency tree · two levels

16 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

Buhler--Salamon, Functional Analysis, Corollary 2.11 (standard reference, not scraped)