TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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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.

[A1]

The stated Dependent Choice is the direct premise for the two open-mapping lemmas (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], The closure of a bounded image contains a ball gives some $r > 0$ with $B_{Y} (0, r) \subseteq \overline{T (B_{X} (0, 1))}$ . The successive-approximation lemma Successive approximation turns a closure-ball inclusion into an actual preimage then gives $B_{Y} (0, r / 2) \subseteq T (B_{X} (0, 1))$ . Put $ε = r / 2 > 0$ .

2.1step 1.1algebra

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

3.1step 2.1∎

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.

Depends on

Used by

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

Dependency tree · two levels

22 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)