Definition:Generalized Sum/Absolute Net Convergence

Definition
Let $V$ be a Banach space.

Let $\family {v_i}_{i \mathop \in I}$ be an indexed family of elements of $V$.

Then $\ds \sum \set {v_i: i \in I}$ converges absolutely $\ds \sum \set {\norm {v_i}: i \mathop \in I}$ converges.

This nomenclature is appropriate as we have Absolutely Convergent Generalized Sum Converges.