TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 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.

Open mapping theorem

Statement

Assume DC. A surjective bounded linear map $T : X \to Y$ between Banach spaces is open: it maps every open subset of $X$ to an open subset of $Y$ .

Facts & Assumptions

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

Proof

technique · direct

1.1

The preceding successive-approximation lemma gives $ε > 0$ with $B_{Y} (0, ε) \subseteq T (B_{X} (0, 1))$ .

given

2.1

For every $x \in X$ and $a > 0$ , linearity gives $B_{Y} (T x, a ε) \subseteq T (B_{X} (x, a))$ .

step 1.1algebra

3.1

Let $U \subseteq X$ be open and $y = T x \in T (U)$ . Choose $a > 0$ with $B_{X} (x, a) \subseteq U$ . Then step 2.1 gives $B_{Y} (y, a ε) \subseteq T (U)$ . Thus every point of $T (U)$ is interior, so $T (U)$ is open.

step 2.1∎

Depends on

Used by

Quantitative lifting form of the open mapping theorem Corollary
Bounded inverse theorem Theorem

Dependency tree · two levels

8 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, Theorem 2.8 (standard reference, not scraped)