Editing Absolute convergence (section)

=== 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.