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.

Net Convergence

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 if and only if the generalized sum $\displaystyle \sum \left\{{g_n: n \in \N}\right\}$ converges.

Absolute Net Convergence

Let $V$ be a Banach space.

Let $\family {v_i}_{i \mathop \in I}$ be an indexed subset of $V$.

Then $\displaystyle \sum \set {v_i: i \in I}$ converges absolutely if and only if $\displaystyle \sum \set {\norm {v_i}: i \mathop \in I}$ 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.

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Sources

• 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $I.4.11$