# Arzelą-Ascoli theorem

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,*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.

*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.