Editing Absolute convergence (section)

== Sums of more general elements ==

The same definition can be used for series <math display=inline>\sum_{n=0}^{\infty} a_n</math> whose terms <math>a_n</math> are not numbers but rather elements of an arbitrary [[Topological abelian group|abelian topological group]]. In that case, instead of using the [[absolute value]], the definition requires the group to have a [[Norm (mathematics)|norm]], which is a positive real-valued function <math display=inline>\|\cdot\|: G \to \R_+</math> on an abelian group <math>G</math> (written [[Abelian group#Notation|additively]], with identity element 0) such that:
# The norm of the identity element of <math>G</math> is zero: <math>\|0\| = 0.</math>
# For every <math>x \in G,</math> <math>\|x\| = 0</math> implies <math>x = 0.</math>
# For every <math>x \in G,</math> <math>\|-x\| = \|x\|.</math>
# For every <math>x, y \in G,</math> <math>\|x+y\| \leq \|x\| + \|y\|.</math>

In this case, the function <math>d(x,y) = \|x-y\|</math> induces the structure of a [[metric space]] (a type of [[topology]]) on <math>G.</math> 

Then, a <math>G</math>-valued series is absolutely convergent if <math display=inline>\sum_{n=0}^{\infty} \|a_n\| < \infty.</math>

In particular, these statements apply using the norm <math>|x|</math> ([[absolute value]]) in the space of real numbers or complex numbers.

=== In topological vector spaces ===

If <math>X</math> is a [[topological vector space]] (TVS) and <math display=inline>\left(x_\alpha\right)_{\alpha \in A}</math> is a (possibly [[uncountable]]) family in <math>X</math> then this family is '''absolutely summable''' if<ref>{{Schaefer Wolff Topological Vector Spaces|edition=2|pp=179–180}}</ref> 
# <math display=inline>\left(x_\alpha\right)_{\alpha \in A}</math> is '''summable''' in <math>X</math> (that is, if the limit <math display=inline>\lim_{H \in \mathcal{F}(A)} x_H</math> of the [[Net (mathematics)|net]] <math>\left(x_H\right)_{H \in \mathcal{F}(A)}</math> converges in <math>X,</math> where <math>\mathcal{F}(A)</math> is the [[directed set]] of all finite subsets of <math>A</math> directed by inclusion <math>\subseteq</math> and <math display=inline>x_H := \sum_{i \in H} x_i</math>), and 
# for every continuous [[seminorm]] <math>p</math> on <math>X,</math> the family <math display="inline">\left(p \left(x_\alpha\right)\right)_{\alpha \in A}</math> is summable in <math>\R.</math>
If <math>X</math> is a normable space and if <math display=inline>\left(x_\alpha\right)_{\alpha \in A}</math> is an absolutely summable family in <math>X,</math> then necessarily all but a countable collection of <math>x_\alpha</math>'s are 0.

Absolutely summable families play an important role in the theory of [[nuclear space]]s.