Definition:Generalized Sum

Definition
Let $\left({G, +}\right)$ be a 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 $I$, and 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 $\phi \left({F}\right) = \displaystyle \sum_{i \in F} g_i$.

Then one denotes $\displaystyle \sum \left\{{g_i: i \in I}\right\}$ for $\phi$ and calls it a generalized sum.

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

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

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

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

Absolute Net Convergence
Let $V$ be a Banach space.

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

One says that $\displaystyle \sum \left\{{v_i: i \in I}\right\}$ converges absolutely if $\displaystyle \sum \left\{{\left\Vert{v_i}\right\Vert: i \in I}\right\}$ converges.

This nomenclature is appropriate as we have Absolutely Convergent Generalized Sum Converges.

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.