Definition:Locally Uniform Convergence/Series

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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.