Definition:Locally Uniform Convergence/Series

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 the sequence of partial sums converges locally uniformly.