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.