LemmaStatement: 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.

The closure of a bounded image contains a ball

Statement

Assume DC. If $T : X \to Y$ is a surjective bounded linear map between Banach spaces, then for some $r > 0$ ,

$B_{Y} (0, r) \subseteq \overline{T (B_{X} (0, 1))} .$

Facts & Assumptions

Given: DC and a surjective bounded linear map $T : X \to Y$ with $X, Y$ Banach.

[F1]

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

Proof

technique · direct

1.1givenF1

Surjectivity gives $Y = ⋃_{n \geq 1} \overline{T (B_{X} (0, n))}$ . Baire applied to $Y$ under the stated DC premise gives $n$ , $y_{0}$ , and $ρ > 0$ with $B (y_{0}, ρ)$ inside this closure.

2.1step 1.1algebra

Subtracting two points in this ball shows $B (0, ρ) \subseteq \overline{T (B (0, 2 n))}$ .

3.1step 2.1algebra∎

Scaling by $2 n$ gives $B (0, ρ / (2 n)) \subseteq \overline{T (B (0, 1))}$ , as required.

Depends on

Dependency tree · 2 levels

Sources

Buhler--Salamon, Functional Analysis, Lemma 2.9