Banach–Mazur theorem
E421069
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Banach–Mazur theorem canonical | 2 |
| Banach–Mazur theorem on bases | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4219689 — 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–Mazur theorem Context triple: [Stefan Banach, eponymOf, Banach–Mazur 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.
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.
-
C.
Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
-
D.
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).
-
E.
Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
- 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–Mazur theorem Target entity description: The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
-
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.
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.
-
C.
Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
-
D.
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).
-
E.
Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in functional analysis ⓘ |
| assumes |
Banach space is complete
ⓘ
Banach space is separable ⓘ |
| codomain | C(K) for some compact metric space K ⓘ |
| concerns |
closed subspaces
ⓘ
compact metric spaces ⓘ isometric isomorphisms ⓘ separable Banach spaces ⓘ spaces of continuous functions ⓘ |
| conclusion |
existence of an isometric linear embedding into C(K)
ⓘ
image of the embedding is a closed subspace of C(K) ⓘ |
| domain | separable Banach space ⓘ |
| field |
Banach space theory
ⓘ
functional analysis ⓘ |
| hasImportance |
allows reduction of problems about separable Banach spaces to problems about C(K) spaces
ⓘ
provides a canonical model for separable Banach spaces ⓘ |
| hasProofTechnique |
functional analytic methods
ⓘ
use of dual spaces and evaluation maps ⓘ |
| holdsFor |
complex separable Banach spaces
ⓘ
real separable Banach spaces ⓘ |
| implies | separable Banach spaces can be represented as spaces of continuous functions on compact metric spaces ⓘ |
| isGeneralizationOf | representation of separable Banach spaces as function spaces ⓘ |
| language | mathematical analysis ⓘ |
| namedAfter |
Stanisław Mazur
NERFINISHED
ⓘ
Stefan Banach NERFINISHED ⓘ |
| relatedTo |
Gelfand transform
ⓘ
surface form:
Gelfand representation
Riesz representation theorem ONNED1 ⓘ Stone representation theorem ⓘ
surface form:
Stone representation theorems
|
| relatesTo |
Banach–Mazur compactum
NERFINISHED
ⓘ
Banach–Mazur distance ⓘ |
| statesThat |
every separable Banach space is isometrically isomorphic to a closed subspace of C(K) for some compact metric space K
ⓘ
for every separable Banach space X there exists a compact metric space K and an isometric linear embedding of X into C(K) ⓘ |
| typicalK | compact metric space constructed from the dual unit ball with weak-star topology ⓘ |
| usedIn |
classification of separable Banach spaces up to isometry
ⓘ
embedding problems in Banach space theory ⓘ study of universal Banach spaces ⓘ |
| usesConcept |
Banach space
ⓘ
C(K) space ⓘ closed subspace ⓘ compactness ⓘ continuous function ⓘ isometry ⓘ linear isometry ⓘ metric space ⓘ norm ⓘ separability ⓘ |
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–Mazur theorem Description of subject: The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Banach–Mazur theorem on bases