Compact convergence

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compact-open topology.

Definition

${\displaystyle f_{n}:X\to Y}$, ${\displaystyle n\in {\mathbb {N} },}$

is said to converge compactly as ${\displaystyle n\to \infty }$ to some function ${\displaystyle f:X\to Y}$ if, for every compact set ${\displaystyle K\subseteq X}$,

${\displaystyle (f_{n})|_{K}\to f|_{K}}$
${\displaystyle \lim _{n\to \infty }\sup _{x\in K}d_{Y}\left(f_{n}(x),f(x)\right)=0.}$

Examples

• A very powerful tool for showing compact convergence is the Arzelà–Ascoli theorem. There are several versions of this theorem, roughly speaking it states that every sequence of equicontinuous and uniformly bounded maps has a subsequence which converges compactly to some continuous map.