Definition:Generalized Sum

Definition
Let $\struct {G, +}$ be a commutative topological semigroup.

Let $\family {g_i}_{i \mathop \in I}$ be an indexed family of elements of $G$.

Consider the set $\FF$ of finite subsets of $I$.

Let $\subseteq$ denote the subset relation on $\FF$.

By virtue of Finite Subsets form Directed Set, $\struct {\FF, \subseteq}$ is a directed set.

Define the net:
 * $\phi: \FF \to G$

by:
 * $\ds \map \phi F = \sum_{i \mathop \in F} g_i$

where $\ds \sum_{i \mathop \in F} g_i$ denotes the summation over $F \in \FF$.

Then $\phi$ is denoted:
 * $\ds \sum \set {g_i: i \in I}$

and referred to as a generalized sum.

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

Also presented as
The generalized sum $\ds \sum \set {g_i: i \in I}$ can also be presented as:


 * $\ds \sum_{i \mathop \in I} \set {g_i}$

The interpretation is obvious.

Motivation
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.