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) $I.4.11$