About: Arzelà–Ascoli theorem

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The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.

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rdfs:label	Arzelà–Ascoli theorem (en)
rdfs:comment	The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators. (en)
sameAs	Arzelà–Ascoli theorem
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Subject	Articles containing proofs Continuous mappings Theorems in functional analysis Topology of function spaces Compactness theorems
Link from a Wikipage to an external page	https://zenodo.org/record/1428464/files/article.pdf http://www.encyclopediaofmath.org/index.php%3Ftitle=Arzel%C3%A0-Ascoli_theorem&oldid=34445
gold:hypernym	Result
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Arzelà–Ascoli_theorem?oldid=1073640290&ns=0
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Wikipage revision ID	1073640290 (xsd:integer)
Link from a Wikipage to another Wikipage	Uniform boundedness Uniform convergence Uniform norm Uniform space Uniformly Cauchy sequence Banach space Bolzano–Weierstrass theorem Montel's theorem Separable space Sequence Bounded set Cantor's diagonal argument Articles containing proofs Continuous mappings Theorems in functional analysis Topology of function spaces Cauchy's integral formula Giulio Ascoli Differentiable function Dual space Metric space Peter–Weyl theorem Cesare Arzelà Hölder's inequality Hölder condition Linear map Lipschitz continuity Springer Science+Business Media Closed set Compact-open topology Compact operator Compact space Compactly generated space Complex analysis Continuous function Continuous functions on a compact Hausdorff space Cover (topology) Holomorph Ordinary differential equation Oscillation (mathematics) Partial differential equation Sobolev space Equicontinuity Fréchet–Kolmogorov theorem Derivative Harmonic analysis Hausdorff space Heine–Borel theorem Helly's selection theorem Infimum and supremum Interval (mathematics) Peano existence theorem Compactness theorems Euclidean space Absolutely integrable function Subsequence Topological vector space Numerical analysis Mean value theorem Necessity and sufficiency Rational number Rational point Real line Real number Relatively compact subspace
has abstract	The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators. The notion of equicontinuity was introduced in the late 19th century by the Italian mathematicians Cesare Arzelà and Giulio Ascoli. A weak form of the theorem was proven by , who established the sufficient condition for compactness, and by , who established the necessary condition and gave the first clear presentation of the result. A further generalization of the theorem was proven by , to sets of real-valued continuous functions with domain a compact metric space . Modern formulations of the theorem allow for the domain to be compact Hausdorff and for the range to be an arbitrary metric space. More general formulations of the theorem exist that give necessary and sufficient conditions for a family of functions from a compactly generated Hausdorff space into a uniform space to be compact in the compact-open topology; see , page 234). (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Arzelà–Ascoli_theorem
is Wikipage redirect of	Arzela-Ascoli Arzela-Ascoli theorem Arzela–Ascoli theorem Arzelà-Ascoli Arzelà-Ascoli theorem

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