Editing Arzelà–Ascoli theorem (section)

{{short description|On when a family of real, continuous functions has a uniformly convergent subsequence}}
The '''Arzelà–Ascoli theorem''' is a fundamental result of [[mathematical analysis]] giving [[necessary and sufficient conditions]] to decide whether every [[sequence (mathematics)|sequence]] of a given family of [[real number|real]]-valued [[continuous function]]s defined on a [[Closed set|closed]] and [[Bounded set|bounded]] [[Interval (mathematics)|interval]] has a [[Uniform convergence|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 operator]]s.

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 {{harvtxt|Ascoli|1883–1884}}, who established the sufficient condition for compactness, and by {{harvtxt|Arzelà|1895}}, who established the necessary condition and gave the first clear presentation of the result.  A further generalization of the theorem was proven by {{harvtxt|Fréchet|1906}}, to sets of real-valued continuous functions with domain a [[Compact space|compact]] [[metric space]] {{harv|Dunford|Schwartz|1958|p=382}}.  Modern formulations of the theorem allow for the domain to be compact [[Hausdorff space|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 space|compactly generated]] Hausdorff space into a [[uniform space]] to be compact in the [[compact-open topology]]; see {{harvtxt|Kelley|1991|loc=page 234}}.