Definition:Generalized Sum/Net Convergence

Definition
Let $\left({G, +}\right)$ be a commutative topological semigroup.

Let $\left({g_n}\right)_{n \in \N}$ be a sequence in $G$.

The series $\displaystyle \sum_{n \mathop = 1}^\infty g_n$ converges as a net or has net convergence the generalized sum $\displaystyle \sum \left\{{g_n: n \in \N}\right\}$ converges.

Also see

 * Net Convergence Equivalent to Absolute Convergence: when $G$ is a Banach space, net convergence is equivalent to absolute convergence.