Arzelà–Ascoli theorem
E898497
The Arzelà–Ascoli theorem is a fundamental result in analysis that characterizes the relative compactness of families of functions via uniform boundedness and equicontinuity.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Arzelà–Ascoli theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991920 — 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: Arzelà–Ascoli theorem Context triple: [Montel's theorem, usesConcept, Arzelà–Ascoli theorem]
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A.
Bolzano–Weierstrass theorem
The Bolzano–Weierstrass theorem is a fundamental result in real analysis stating that every bounded infinite sequence in ℝⁿ has a convergent subsequence.
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B.
Stone–Weierstrass theorem
The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.
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C.
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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D.
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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E.
Rellich–Kondrachov compactness theorem
The Rellich–Kondrachov compactness theorem is a fundamental result in functional analysis and the theory of Sobolev spaces that guarantees the compactness of certain embedding operators, playing a key role in the study of partial differential equations.
- 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: Arzelà–Ascoli theorem Target entity description: The Arzelà–Ascoli theorem is a fundamental result in analysis that characterizes the relative compactness of families of functions via uniform boundedness and equicontinuity.
-
A.
Bolzano–Weierstrass theorem
The Bolzano–Weierstrass theorem is a fundamental result in real analysis stating that every bounded infinite sequence in ℝⁿ has a convergent subsequence.
-
B.
Stone–Weierstrass theorem
The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.
-
C.
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.
-
D.
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.
-
E.
Rellich–Kondrachov compactness theorem
The Rellich–Kondrachov compactness theorem is a fundamental result in functional analysis and the theory of Sobolev spaces that guarantees the compactness of certain embedding operators, playing a key role in the study of partial differential equations.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
compactness theorem
ⓘ
theorem in mathematical analysis ⓘ |
| appliesTo |
families of complex-valued continuous functions
ⓘ
families of real-valued continuous functions ⓘ functions defined on compact Hausdorff spaces ⓘ functions defined on compact metric spaces ⓘ |
| assumption |
domain is compact or locally compact with suitable modifications
ⓘ
family of functions is equicontinuous ⓘ family of functions is uniformly bounded ⓘ |
| characterizes |
precompact subsets of C(K)
ⓘ
relative compactness of families of functions ⓘ |
| codomainCondition |
codomain is a metric space
ⓘ
codomain is ℂ ⓘ codomain is ℝ ⓘ |
| conclusion |
a family is relatively compact in C(K) iff it is equicontinuous and pointwise relatively compact
ⓘ
every sequence in the family has a uniformly convergent subsequence ⓘ on compact domains, pointwise boundedness plus equicontinuity implies relative compactness in the uniform norm ⓘ |
| domainCondition |
domain is a compact metric space
ⓘ
domain is compact ⓘ |
| field |
functional analysis
ⓘ
mathematical analysis ⓘ topology ⓘ |
| generalizationOf | Bolzano–Weierstrass theorem for functions NERFINISHED ⓘ |
| historicalPeriod | late 19th century ⓘ |
| importance | fundamental tool in analysis for extracting convergent subsequences of functions ⓘ |
| namedAfter |
Cesare Arzelà
NERFINISHED
ⓘ
Giulio Ascoli NERFINISHED ⓘ |
| norm | supremum norm ⓘ |
| relatedTo |
Ascoli theorem
NERFINISHED
ⓘ
Banach–Alaoglu theorem NERFINISHED ⓘ Heine–Cantor theorem NERFINISHED ⓘ Riesz representation theorem NERFINISHED ⓘ |
| resultType |
compactness criterion
ⓘ
sequential compactness criterion ⓘ |
| space | space of continuous functions C(K) ⓘ |
| topology | topology of uniform convergence ⓘ |
| typicalFormulation |
every equicontinuous, uniformly bounded sequence of functions on a compact set has a uniformly convergent subsequence
ⓘ
subset of C(K) is relatively compact iff it is bounded and equicontinuous ⓘ |
| usedIn |
approximation theory
ⓘ
compactness arguments in PDE theory ⓘ existence proofs for integral equations ⓘ existence proofs for solutions of differential equations ⓘ functional analysis of operator families ⓘ |
| usesConcept |
compactness in function spaces
ⓘ
equicontinuity ⓘ relative compactness ⓘ uniform boundedness ⓘ uniform convergence ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Arzelà–Ascoli theorem Description of subject: The Arzelà–Ascoli theorem is a fundamental result in analysis that characterizes the relative compactness of families of functions via uniform boundedness and equicontinuity.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Montel's theorem