Definition:Generalized Sum

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Definition

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

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

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

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

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


Define the net:

$\phi: \FF \to G$

by:

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


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.


Net Convergence

Let $\sequence {g_n}_{n \mathop \in \N}$ be a sequence in $G$.

The series $\ds \sum_{n \mathop = 1}^\infty g_n$ converges as a net or has net convergence if and only if the generalized sum $\ds \sum \set {g_n: n \in \N}$ 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 $\ds \sum \set {v_i: i \in I}$ converges absolutely if and only if $\ds \sum \set {\norm {v_i}: i \mathop \in I}$ converges.

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


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.



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Sources