Definition:Compact Absolute Convergence

Definition
Let $X$ be a topological space.

Let $\left({V, \left\lVert{\, \cdot \,}\right\rVert}\right)$ 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 compactly (uniformly) absolutely the series $\displaystyle \sum_{n \mathop = 1}^\infty \left\lVert{f_n}\right\rVert$ converges compactly.

Also see

 * Definition:Locally Uniform Absolute Convergence
 * Equivalence of Local Uniform Convergence and Compact Convergence