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.
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 |
bounded inverse theorem
ⓘ
linked to:
Banach inverse mapping 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 ⓘ
linked to:
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Banach inverse mapping theorem Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Banach spaces
linked to: Banach inverse mapping theorem
linked to: Banach inverse mapping theorem
linked to: Banach inverse mapping theorem