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In mathematics, the Arzelą-Ascoli theorem of functional analysis is a criterion to decide whether a set of continuous functions from a compact metric space into a metric space is compact in the topology of uniform convergence.
- Arzelą-Ascoli theorem: Let X be a compact metric space, Y a metric space. Then a subset F of C(X, Y) is compact if and only if it is equicontinuous, pointwise relatively compact and closed.
Here, C(X, Y) denotes the set of all continuous functions from X to Y, and a subset F is pointwise relatively compact iff for all x in X, the set {f(x) : f in F} is relatively compact in Y.
The name comes about as this is a generalisation going back to Cesare Arzelą of Ascoli's theorem.