Definition:Generalized Sum

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

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

Consider the set $\mathcal F$ of finite subsets of $I$.

Let $\subseteq$ denote the subset relation on $\mathcal F$.

By virtue of Finite Subsets form Directed Set, $\left({\mathcal F, \subseteq}\right)$ is a directed set.

Define the net:
 * $\phi: \mathcal F \to G$

by:
 * $\displaystyle \phi \left({F}\right) = \sum_{i \mathop \in F} g_i$

Then $\phi$ is denoted:
 * $\displaystyle \sum \left\{{g_i: i \in I}\right\}$

and referred to as a generalized sum.

Statements about convergence of $\displaystyle \sum \left\{{g_i: i \in I}\right\}$ are as for general convergent nets.

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.