Characterizes when a family of continuous functions on a compact space has a uniformly convergent subsequence, in terms of the family being uniformly bounded and equicontinuous. It is a central compactness criterion in functional analysis, named for Cesare Arzela and Giulio Ascoli.
Facts
StatementIf a sequence of real-valued continuous functions on a closed and bounded interval is uniformly bounded and uniformly equicontinuous, then it has a subsequence that converges uniformly. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Sources
1. Arzela-Ascoli Theorem (Wikipedia)
Wikimedia FoundationStatement and first consequences section
Consider a sequence of real-valued continuous functions f_n defined on a closed and bounded interval [a, b] of the real line. If this sequence is uniformly bounded and uniformly equicontinuous, then there exists a subsequence that converges uniformly.
lead section, history sentence
A weak form of the theorem was proven by Ascoli (1883-1884), who established the sufficient condition for compactness, and by Arzelà (1895), who established the necessary condition and gave the first clear presentation of the result.
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