Banach–Saks theorem
E421068
The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Banach–Saks property | 1 |
| Banach–Saks theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4219688 — 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–Saks theorem Context triple: [Stefan Banach, eponymOf, Banach–Saks theorem]
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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.
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C.
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.
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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.
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E.
Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
- 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–Saks theorem Target entity description: The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
-
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.
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.
-
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.
Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
- F. None of above. chosen
Statements (37)
| Predicate | Object |
|---|---|
| instanceOf |
result in Banach space theory
ⓘ
theorem in functional analysis ⓘ |
| appliesTo |
Hilbert spaces
ⓘ
L^p spaces for 1 < p < ∞ ⓘ |
| assumption |
The sequence considered is bounded.
ⓘ
The underlying space is a reflexive Banach space. ⓘ |
| characterizes | a property of bounded sequences in reflexive Banach spaces ⓘ |
| concerns |
Cesàro means of sequences
ⓘ
bounded sequences in Banach spaces ⓘ norm convergence ⓘ reflexive Banach spaces ⓘ |
| conclusion | There exists a subsequence whose Cesàro means converge in norm. ⓘ |
| contrastWith | behavior of bounded sequences in non-reflexive Banach spaces ⓘ |
| doesNotRequire | the original sequence to be convergent ⓘ |
| field |
Banach space theory
ONNED1
ⓘ
functional analysis ⓘ |
| formalizes | improvement from weak to strong convergence for Cesàro means ⓘ |
| guarantees | existence of a norm-convergent sequence of Cesàro means ⓘ |
| historicalContext | proved in the early development of Polish functional analysis ⓘ |
| implies | reflexive Banach spaces have the Banach–Saks property ⓘ |
| isStrongerThan | results that only give weak convergence of subsequences ⓘ |
| namedAfter |
Stanisław Saks
NERFINISHED
ⓘ
Stefan Banach NERFINISHED ⓘ |
| relatedConcept |
Banach–Saks theorem
self-linksurface differs
ⓘ
surface form:
Banach–Saks property
Cesàro convergence ⓘ reflexivity of Banach spaces ⓘ weak convergence ⓘ |
| statement | Every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm. ⓘ |
| typeOfResult | compactness-type theorem in Banach spaces ⓘ |
| typicalProofUses |
diagonal subsequence arguments
ⓘ
properties of Cesàro averages ⓘ weak compactness in reflexive spaces ⓘ |
| usedIn |
ergodic-type averaging arguments in analysis
ⓘ
study of structure of reflexive Banach spaces ⓘ |
| usesConcept |
bounded set in a Banach space
ⓘ
norm topology ⓘ subsequence ⓘ |
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–Saks theorem Description of subject: The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Banach–Saks property