Definition:Locally Uniform Convergence/Series
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Definition
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.