Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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 closure of a bounded image contains a ball

Statement

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

BY(0,r)⊆T(BX(0,1))‾.

Facts & Assumptions

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

Proof

technique · direct
1.1

Surjectivity gives Y=⋃n≥1T(BX(0,n))‾. Baire applied to Y gives n, y0, and ρ>0 with B(y0,ρ) inside this closure.

given
2.1

Subtracting two points in this ball shows B(0,ρ)⊆T(B(0,2n))‾.

step 1.1algebra
3.1

Scaling by 2n gives B(0,ρ/(2n))⊆T(B(0,1))‾, as required.

step 2.1algebra∎

Depends on

Used by

Dependency tree · two levels

13 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