Riesz lemma
E746577
Riesz lemma is a fundamental result in functional analysis that characterizes how, in an infinite-dimensional normed space, one can find unit vectors that stay a fixed distance away from any given proper closed subspace.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Riesz lemma canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T8640753 — 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: Riesz lemma Context triple: [Frigyes Riesz, knownFor, Riesz lemma]
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A.
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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B.
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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C.
Banach–Mazur theorem
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.
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D.
Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
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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: Riesz lemma Target entity description: Riesz lemma is a fundamental result in functional analysis that characterizes how, in an infinite-dimensional normed space, one can find unit vectors that stay a fixed distance away from any given proper closed subspace.
-
A.
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.
-
B.
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.
-
C.
Banach–Mazur theorem
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.
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D.
Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
-
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 (40)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in functional analysis ⓘ |
| appliesTo |
Banach spaces
ⓘ
normed linear spaces ⓘ normed vector spaces ⓘ |
| assumption |
subspace is closed
ⓘ
subspace is proper ⓘ |
| conclusion | there exists a unit vector whose distance from the subspace is at least α ⓘ |
| domainCondition |
infinite-dimensional normed space
ⓘ
proper closed subspace ⓘ |
| field |
functional analysis
ⓘ
normed vector spaces ⓘ |
| guaranteesExistenceOf | unit vector at prescribed distance from a proper closed subspace ⓘ |
| holdsIn |
complex normed spaces
ⓘ
real normed spaces ⓘ |
| implies |
in an infinite-dimensional normed space, the closed unit ball is not compact in the norm topology
ⓘ
in an infinite-dimensional normed space, the closed unit ball is not sequentially compact ⓘ |
| involvesConcept |
closed subspace
ⓘ
distance to a subspace ⓘ infinite-dimensionality ⓘ norm ⓘ unit vector ⓘ |
| mathematicsSubjectClassification |
46Axx
ⓘ
46Bxx ⓘ |
| namedAfter | Frigyes Riesz ⓘ |
| parameter | real number α with 0 < α < 1 ⓘ |
| relatedTo |
Banach–Alaoglu theorem
ⓘ
Hahn–Banach theorem ⓘ Riesz representation theorem ⓘ geometric theory of Banach spaces ⓘ |
| statement | If X is a normed space, Y is a proper closed subspace of X, and 0 < α < 1, then there exists x in X with ∥x∥ = 1 such that the distance from x to Y is at least α. ⓘ |
| strengthenedBy | various quantitative versions in Banach space theory ⓘ |
| type | existence lemma ⓘ |
| usedFor |
constructing basic sequences in Banach spaces
ⓘ
constructing sequences without convergent subsequences in infinite-dimensional spaces ⓘ demonstrating geometric properties of normed spaces ⓘ proving that closed unit balls in infinite-dimensional normed spaces are not compact ⓘ showing non-compactness of the unit ball in infinite-dimensional Banach spaces ⓘ |
| usedInProofOf |
characterizations of finite-dimensional normed spaces via compactness of the unit ball
ⓘ
results on basic sequences and Schauder bases ⓘ |
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: Riesz lemma Description of subject: Riesz lemma is a fundamental result in functional analysis that characterizes how, in an infinite-dimensional normed space, one can find unit vectors that stay a fixed distance away from any given proper closed subspace.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.