# Definition:Locally Uniform Convergence

## Definition

### General Definition

Let $X$ be a topological space.

Let $M$ be a metric space.

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

Then $f_n$ converges locally uniformly to $f: X \to M$ 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 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 $r > 0$ 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 $\left\langle{f_n}\right\rangle$ be a sequence of mappings $f_n:X\to V$.

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