Category:Generalized Sums

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This category contains results about Generalized Sums.
Definitions specific to this category can be found in Definitions/Generalized Sums.


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.