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 $z$.
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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 $\sequence {f_n}$ 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.