Banach inverse mapping theorem

E121357

result in operator theory theorem in functional analysis theorem in functional analysis of Banach spaces

The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.

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All labels observed (4)

Label	Occurrences
Banach inverse mapping theorem canonical	1
Bounded Inverse Theorem	1
Open Mapping Theorem	1
bounded inverse theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T1057115 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Banach inverse mapping theorem
Context triple: [inverse function theorem, isRelatedTo, Banach inverse mapping theorem]

A. inverse function theorem
The inverse function theorem is a fundamental result in calculus and differential geometry that gives conditions under which a differentiable function has a locally defined differentiable inverse near a point where its derivative is invertible.
B. Banach spaces
Banach spaces are complete normed vector spaces that provide a fundamental framework for functional analysis and the study of infinite-dimensional linear phenomena.
C. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
D. Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
E. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Banach inverse mapping theorem
Target entity description: The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.

A. inverse function theorem
The inverse function theorem is a fundamental result in calculus and differential geometry that gives conditions under which a differentiable function has a locally defined differentiable inverse near a point where its derivative is invertible.
B. Banach spaces
Banach spaces are complete normed vector spaces that provide a fundamental framework for functional analysis and the study of infinite-dimensional linear phenomena.
C. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
D. Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
E. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	result in operator theory ⓘ theorem in functional analysis ⓘ theorem in functional analysis of Banach spaces ⓘ
appearsIn	Stefan Banach’s work on linear operations ⓘ
appliesTo	Banach spaces ⓘ bounded linear operators ⓘ
assumes	codomain is a Banach space ⓘ domain is a Banach space ⓘ operator is bijective ⓘ operator is bounded ⓘ operator is linear ⓘ
categoryTheoreticView	characterizes isomorphisms in the category of Banach spaces and bounded linear maps ⓘ
classification	fundamental theorem of Banach space theory ⓘ
concludes	inverse operator exists ⓘ inverse operator is bounded ⓘ inverse operator is continuous ⓘ inverse operator is linear ⓘ
ensures	bijective bounded linear operator between Banach spaces is a topological isomorphism ⓘ graph of inverse operator is closed ⓘ
equivalentTo	Banach inverse mapping theorem self-linksurface differs ⓘ surface form: bounded inverse theorem
field	functional analysis ⓘ mathematics ⓘ
generalizationOf	inverse function theorem for linear maps between finite-dimensional normed spaces ⓘ
hasConsequence	closed range of a bijective bounded operator is the whole codomain ⓘ norms on a finite-dimensional vector space are equivalent ⓘ
holdsIn	complex Banach spaces ⓘ real Banach spaces ⓘ
implies	homeomorphism between Banach spaces via bounded linear bijection ⓘ open mapping theorem ⓘ
namedAfter	Stefan Banach ⓘ
proofTechnique	Baire category theorem ⓘ Neumann series for operators ⓘ
relatedConcept	bounded linear isomorphism ⓘ topological isomorphism of Banach spaces ⓘ
relatedTo	Hahn–Banach theorem ⓘ Closed Graph Theorem ⓘ surface form: closed graph theorem
requires	completeness of codomain space ⓘ completeness of domain space ⓘ
statedAs	If T:X→Y is a bijective bounded linear operator between Banach spaces, then T^{-1}:Y→X is bounded and linear ⓘ
topicOf	many graduate-level functional analysis courses ⓘ
usedIn	nonlinear functional analysis via linearization ⓘ partial differential equations ⓘ perturbation theory of linear operators ⓘ spectral theory of bounded operators ⓘ study of isomorphisms between Banach spaces ⓘ
usedToShow	closed range of a bounded bijective operator is automatically all of the codomain ⓘ stability of solutions of linear operator equations under perturbations ⓘ

How these facts were elicited

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

inverse function theorem → isRelatedTo → Banach inverse mapping theorem ⓘ

Banach spaces → hasConcept → Banach inverse mapping theorem ⓘ

subject surface form: Banach space

this entity surface form: Open Mapping Theorem

Banach inverse mapping theorem → equivalentTo → Banach inverse mapping theorem self-linksurface differs ⓘ

this entity surface form: bounded inverse theorem

Closed Graph Theorem → relatedTo → Banach inverse mapping theorem ⓘ

this entity surface form: Bounded Inverse Theorem

All labels observed (4)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (4)

How this entity was disambiguated

How these facts were elicited