Category:Definitions/Generalized Sums

This category contains definitions related to Generalized Sums.
Related results can be found in Category:Generalized Sums.

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.