Banach–Stone theorem
E421067
The Banach–Stone theorem is a fundamental result in functional analysis that characterizes compact Hausdorff spaces via isometric isomorphisms between their spaces of continuous real- or complex-valued functions.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Banach–Stone theorem canonical | 1 |
| real Banach–Stone theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4219687 — 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–Stone theorem Context triple: [Stefan Banach, eponymOf, Banach–Stone theorem]
-
A.
Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
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B.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
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C.
Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
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D.
Banach inverse mapping theorem
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.
-
E.
Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
- 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–Stone theorem Target entity description: The Banach–Stone theorem is a fundamental result in functional analysis that characterizes compact Hausdorff spaces via isometric isomorphisms between their spaces of continuous real- or complex-valued functions.
-
A.
Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
-
B.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
-
C.
Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
-
D.
Banach inverse mapping theorem
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.
-
E.
Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in functional analysis ⓘ |
| appliesTo |
C(X)
ⓘ
C(Y) ⓘ |
| assumes | surjective linear isometry between C(X) and C(Y) ⓘ |
| characterizes |
compact Hausdorff spaces up to homeomorphism
ⓘ
isometric isomorphisms between C(X) and C(Y) ⓘ |
| codomainCondition |
C(X) Banach space with sup norm
ⓘ
C(Y) Banach space with sup norm ⓘ |
| concerns |
compact Hausdorff spaces
ⓘ
isometric isomorphisms of Banach spaces ⓘ spaces of continuous functions ⓘ |
| concludes |
existence of homeomorphism φ:Y→X
ⓘ
representation of isometry via composition with φ and multiplication by unimodular function ⓘ |
| domainCondition |
X compact Hausdorff space
ⓘ
Y compact Hausdorff space ⓘ |
| field |
functional analysis
ⓘ
topology ⓘ |
| hasGeneralization |
Banach–Stone type theorems for C0(X)
ⓘ
Banach–Stone type theorems for Lipschitz function spaces ⓘ Banach–Stone type theorems for vector-valued function spaces ⓘ Gelfand–Naimark theorem ⓘ |
| hasVariant |
complex Banach–Stone theorem
ONNED1
ⓘ
Banach–Stone theorem self-linksurface differs ⓘ
surface form:
real Banach–Stone theorem
|
| holdsFor |
continuous complex-valued functions on compact Hausdorff spaces
ⓘ
continuous real-valued functions on compact Hausdorff spaces ⓘ |
| implies |
isometric isomorphism class of C(X) determines X up to homeomorphism
ⓘ
topological structure of X is determined by Banach space structure of C(X) ⓘ |
| mathematicsSubjectClassification |
46E15
ⓘ
54C35 ⓘ |
| namedAfter |
Marshall Harvey Stone
NERFINISHED
ⓘ
Stefan Banach NERFINISHED ⓘ |
| relatedTo |
C(K) spaces
ⓘ
Gelfand representation of commutative C*-algebras ⓘ Riesz representation theorem ⓘ |
| statement |
Every surjective linear isometry T:C(X)→C(Y) is induced by a homeomorphism between X and Y and a unimodular function
ⓘ
If C(X) and C(Y) are isometrically isomorphic as Banach spaces then X and Y are homeomorphic ⓘ |
| topic |
duality between topology and function spaces
ⓘ
isometric classification of C(K) spaces ⓘ |
| uses |
homeomorphism
ⓘ
linear isometry ⓘ supremum norm ⓘ |
| valueType |
complex-valued continuous functions
ⓘ
real-valued continuous functions ⓘ |
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–Stone theorem Description of subject: The Banach–Stone theorem is a fundamental result in functional analysis that characterizes compact Hausdorff spaces via isometric isomorphisms between their spaces of continuous real- or complex-valued functions.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
real Banach–Stone theorem