The Arzelà-Ascoli theorem, also called Ascoli's theorem, characterizes compact closures of families
of continuous functions. Let 
be a nonempty compact metric space and let 
be a family of continuous
functions 
.
The set closure of 
in the supremum norm
 |
is compact iff 
is equicontinuous and there
is a constant 
such that
 |
for every 
and 
.
In particular, every sequence in such a family
has a uniformly convergent subsequence
whose limit is a continuous
function (Rudin 1991, p. 394).
