TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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.

Proof

technique · direct

1.1

By Open mapping theorem, $T$ maps the open unit ball onto a neighbourhood of $0$ ; hence $B_{Y} (0, c) \subseteq T (B_{X} (0, 1))$ for some $c > 0$ .

given

2.1

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 $0$ .

step 1.1algebra

3.1

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

step 2.1∎

Depends on

Used by

A one-sided comparison of two complete norms makes them equivalent Corollary
Closed graph theorem Theorem

Dependency tree · two levels

7 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 2.12 (standard reference, not scraped)