Definition:Generalized Sum

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

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

Consider the set $F$ of finite subsets of $I$, and let $\le$ denote the subset relation on $F$.

By virtue of Subset Relation on Power Set is Partial Ordering, $\left({F, \le}\right)$ is a poset.

Define the net $\phi: F \to G$ by $\phi(S) = \sum_{i \in S} g_i$.

Then the generalized sum $\sum \left\{g_i: i \in I\right\}$ is defined as the limit of the net $\phi$.

If the limit does not exist, the sum is said to diverge.

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

One says that the series $\sum_{n=1}^\infty g_n$ converges as a net or has net convergence if $\sum \left\{g_n: n \in \N\right\}$ converges.

By Net Convergence Equivalent to Absolute Convergence, when $G$ is a normed vector space, this is equivalent to absolute convergence.

Note
While the notion of a topological group may be somewhat overwhelming, one may as well read normed vector space in its place to at least grasp the most important use of a generalized sum.