Definition:Generalized Sum/Absolute Net Convergence

Definition
Let $V$ be a Banach space.

Let $\left({v_i}\right)_{i \in I}$ be an indexed subset of $V$.

Then $\displaystyle \sum \left\{{v_i: i \in I}\right\}$ converges absolutely iff $\displaystyle \sum \left\{{\left\Vert{v_i}\right\Vert: i \in I}\right\}$ converges.

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