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Bounded inverse theorem
Statement
Assume DC. A bounded bijective linear map between Banach spaces has a bounded linear inverse .
Facts & Assumptions
Given: DC and a bounded bijective linear between Banach spaces.
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 -indexed chain).
Proof
Under [A1], Open mapping theorem makes an open neighbourhood of ; hence for some .
If , the point lies in , so its unique preimage has norm . Scaling gives ; this also holds at .
Thus is bounded, and it is linear because is a linear bijection.
Depends on
Used by
- A one-sided comparison of two complete norms makes them equivalent Corollary
- Invertibility and the inverse of the transpose Corollary
- Non-invertible elliptic shifts form a discrete set in the self-adjoint case Corollary
- Normal operator norm equals spectral radius Corollary
- Uniqueness implies existence for the elliptic Dirichlet problem Corollary
- Point continuous and residual spectrum Definition
- Baire diagonal passage from finite regularity to smooth metrics Lemma
- Character space of generated normal algebra is operator spectrum Lemma
- Closed range is equivalent to a quotient estimate Lemma
- Fredholm splitting and parametrix Lemma
- Local finite-dimensional reduction for a Fredholm map Lemma
- Riesz Schauder ascent and descent stabilize Lemma
- The universal metric--trajectory projection is Fredholm Lemma
- Bounded normal operator abstract spectral theorem Theorem
- Closed graph theorem Theorem
- Continuous functional calculus for bounded self adjoint operators Theorem
- Coordinate functionals of a Schauder basis are bounded Theorem
- Fredholm alternative for identity minus compact Theorem
- Regular value theorem for Banach manifolds Theorem
- Spectral theorem for bounded normal operators pvm form Theorem
Dependency tree · two levels
20 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)