Uniform convergence on compact sets

Uniform convergence on every compact subset of the domain.

Uniform convergence on compact sets

A sequence of functions $(f_n):X\to Y$ (with $Y$ a metric space ) converges uniformly on compact sets to a function $f:X\to Y$ if for every compact subset $K\subseteq X$ , the restricted sequence $f_n|_K$ converges uniformly to $f|_K$ on $K$ , i.e.

\forall \varepsilon>0\ \exists N\ \forall n\ge N:\ \sup_{x\in K} d\bigl(f_n(x),f(x)\bigr)<\varepsilon.

This is a localized version of uniform convergence obtained by first passing to a restriction on each compact set. It is the natural convergence notion for families like power series , as formalized by uniform convergence on compacts for power series .

Examples:

On $(-1,1)$ , $f_n(x)=x^n$ converges uniformly on every closed interval $[-a,a]$ with $0<a<1$ , hence uniformly on compact sets in $(-1,1)$ .
If $\sum_{n=0}^\infty a_n (x-x_0)^n$ has radius of convergence $R>0$ , then its partial sums converge uniformly on compact subsets of $(x_0-R,x_0+R)$ .