Definition:Locally Uniform 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 locally uniformly absolutely the series $\ds \sum_{n \mathop = 1}^\infty \norm {f_n}$ converges locally uniformly.

Also see

 * Definition:Compact Absolute Convergence


 * Equivalence of Local Uniform Convergence and Compact Convergence