Definition:Locally Uniform Convergence

Definition

General Definition

Let $\struct {X, \tau}$ be a topological space.

Let $\struct {M, d}$ be a metric space.

Let $\sequence {f_n}$ be a sequence of mappings $f_n: X \to M$.

Then $f_n$ converges locally uniformly to $f: X \to M$ if and only if every point of $X$ has a neighborhood on which $f_n$ converges uniformly to $f$.

Complex Functions

Let $U \subseteq \C$ be an open set.

Let $\sequence {f_n}$ be a be a sequence of functions $f_n : U \to \C$.

For $z \in U$, let $\map {D_r} z$ be the open disk of radius $r$ about $z$.

Then $f_n$ converges to $f$ locally uniformly if and only if:

for each $z \in U$, there is an $r \in \R_{>0}$ such that $f_n$ converges uniformly to $f$ on $\map {D_r} z$

and:

$\map {D_r} z \subseteq U$

Series

Let $X$ be a topological space.

Let $V$ be a normed vector space.

Let $\sequence {f_n}$ be a sequence of mappings $f_n: X \to V$.

Then the series $\ds \sum_{n \mathop = 1}^\infty f_n$ converges locally uniformly if and only if the sequence of partial sums converges locally uniformly.