\begin{definition}[Definition:Generalized Sum/Absolute Net Convergence]
Let $V$ be a Banach space.
Let $\family {v_i}_{i \mathop \in I}$ be an indexed subset of $V$.
Then $\ds \sum \set {v_i: i \in I}$ '''converges absolutely''' {{iff}} $\ds \sum \set {\norm {v_i}: i \mathop \in I}$ converges.
This nomenclature is appropriate as we have Absolutely Convergent Generalized Sum Converges.
Category:Definitions/Group Theory
Category:Definitions/Generalized Sums
\end{definition}