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

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Bounded inverse theorem

Statement

Assume DC. A bounded bijective linear map $T : X \to Y$ between Banach spaces has a bounded linear inverse $T^{- 1} : Y \to X$ .

Facts & Assumptions

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

[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 $0$ ; hence $B_{Y} (0, c) \subseteq T (B_{X} (0, 1))$ for some $c > 0$ .

2.1step 1.1algebra

If $y \neq 0$ , the point $c y / (2 ∥ y ∥)$ lies in $B_{Y} (0, c)$ , so its unique preimage has norm $< 1$ . Scaling gives $∥ T^{- 1} y ∥ < 2 c^{- 1} ∥ y ∥$ ; this also holds at $0$ .

3.1step 2.1∎

Thus $T^{- 1}$ is bounded, and it is linear because $T$ is a linear bijection.

Depends on

Dependency tree · 2 levels

Sources

Buhler--Salamon, Functional Analysis, Theorem 2.12