Definition:Uniform Absolute Convergence

Definition
Let $S$ be a set.

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 $f_n: S \to V$.

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ converges uniformly absolutely the series $\displaystyle \sum_{n \mathop = 1}^\infty \left\lVert{f_n}\right\rVert$ converges uniformly.

Also see

 * Definition:Uniform Convergence
 * Definition:Absolute Convergence
 * Definition:Local Uniform Absolute Convergence
 * Uniform Absolute Convergence Implies Uniform Convergence