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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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 ⓘ

Referenced by (1)

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

Montel theorem → usesConcept → Arzelà–Ascoli theorem ⓘ

subject surface form: Montel's theorem