Definition:Locally Uniform Convergence/Series

Definition
Let $X$ be a topological space.

Let $V$ be a normed vector space.

Let $\left\langle{f_n}\right\rangle$ be a sequence of mappings $f_n:X\to V$.

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ converges locally uniformly the sequence of partial sums converges locally uniformly.