Editing Net (mathematics) (section)

===Cauchy nets===

A Cauchy net generalizes the notion of [[Cauchy sequence]] to nets defined on [[uniform space]]s.<ref name="willard">{{citation|title=General Topology|series=Dover Books on Mathematics|first=Stephen|last=Willard|publisher=Courier Dover Publications|year=2012|isbn=9780486131788|page=260|url=https://books.google.com/books?id=UrsHbOjiR8QC&pg=PA26}}.</ref>

A net <math>x_\bull = \left(x_a\right)_{a \in A}</math> is a {{em|{{visible anchor|Cauchy net}}}} if for every [[Entourage (mathematics)|entourage]] <math>V</math> there exists <math>c \in A</math> such that for all <math>a, b \geq c,</math> <math>\left(x_a, x_b\right)</math> is a member of <math>V.</math><ref name="willard"/><ref>{{citation|title=Introduction to General Topology|first=K. D.|last=Joshi|publisher=New Age International|year=1983|isbn=9780852264447|page=356|url=https://books.google.com/books?id=fvCpXrube5wC&pg=PA356}}.</ref> More generally, in a [[Cauchy space]], a net <math>x_\bull</math> is Cauchy if the filter generated by the net is a [[Cauchy filter]].

A [[topological vector space]] (TVS) is called {{em|[[Complete topological vector space|complete]]}} if every Cauchy net converges to some point. A [[normed space]], which is a special type of topological vector space, is a complete TVS (equivalently, a [[Banach space]]) if and only if every Cauchy sequence converges to some point (a property that is called {{em|sequential completeness}}). Although Cauchy nets are not needed to describe completeness of normed spaces, they are needed to describe completeness of more general (possibly non-[[Normable space|normable]]) topological vector spaces.