# Definition:Compact Absolute Convergence

## Definition

Let $X$ be a topological space.

Let $\struct {V, \norm {\, \cdot \,} }$ 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 compactly (uniformly) absolutely if and only if the series $\ds \sum_{n \mathop = 1}^\infty \norm {f_n}$ converges compactly.