Arzelà–Ascoli theorem

E898497

compactness theorem theorem in mathematical analysis

The Arzelà–Ascoli theorem is a fundamental result in analysis that characterizes the relative compactness of families of functions via uniform boundedness and equicontinuity.

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All labels observed (2)

Label	Occurrences
Arzelà–Ascoli theorem canonical	3
Ascoli theorem	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]

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
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 ⓘ linked to: Bolzano–Weierstrass theorem
historicalPeriod	late 19th century ⓘ
importance	fundamental tool in analysis for extracting convergent subsequences of functions ⓘ
namedAfter	Cesare Arzelà ⓘ Giulio Ascoli NERFINISHED ⓘ
norm	supremum norm ⓘ
relatedTo	Ascoli theorem ⓘ linked to: Arzelà–Ascoli theorem Banach–Alaoglu theorem ⓘ Heine–Cantor theorem ⓘ Riesz representation theorem ⓘ
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

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Montel's theorem → usesConcept → Arzelà–Ascoli theorem ⓘ

subject linked to: Montel theorem

Dini's theorem → relatedTo → Arzelà–Ascoli theorem ⓘ

Cheeger–Gromov compactness theorem → relatedTo → Arzelà–Ascoli theorem ⓘ

Arzelà–Ascoli theorem → relatedTo → Ascoli theorem ⓘ

linked to: Arzelà–Ascoli theorem

All labels observed (2)

How this entity was disambiguated Show

Statements (48)

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Referenced by (4)

How this entity was disambiguated

How these facts were elicited