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 $z$.
Given that "disk" is just another term for "ball", it remains to be specified whether that ball is open or closed.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
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.
