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.

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 family of elements 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.

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.

This article is complete as far as it goes, but it could do with expansion. In particular: Actually, I would like to use the terminology Topological Vector Space for the most general definition, but it is not until Chapter IV of Conway that this will be introduced. So please be patient. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text I.4.11$