Arzelà-Ascoli theorem

From Wikipedia, the free encyclopedia

In mathematics, the Arzelà-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.

In the simplest terms this theorem can be stated as follows. Consider a sequence of continuous functions $(f n)$ defined on a closed interval $[a, b]$ of the real line with real values. If this sequence is uniformly bounded and uniformly equicontinuous, then it admits a subsequence $(f_{n_k})$ which converges uniformly. The assumptions of the theorem are for example satisfied for a sequence of differentiable functions which is uniformly bounded and the sequence of derivatives is also uniformly bounded.

The general version of this theorem is as follows.

Arzelà-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 if and only if for all x in X, the set {f(x) : f in F} is relatively compact in Y.