# Sequentially complete

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Sequentially_complete
> Markdown URL: https://mediated.wiki/source/Sequentially_complete.md
> Source: https://en.wikipedia.org/wiki/Sequentially_complete
> Source revision: 1239328533
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{more footnotes|date=May 2020}}

In mathematics, specifically in [topology](/source/topology) and [functional analysis](/source/functional_analysis), a subspace {{mvar|S}} of a [uniform space](/source/uniform_space) {{mvar|X}} is said to be '''sequentially complete''' or '''semi-complete''' if every [Cauchy sequence](/source/Cauchy_sequence) in {{mvar|S}} converges to an element in {{mvar|S}}. 
{{mvar|X}} is called '''sequentially complete''' if it is a sequentially complete subset of itself. 

== Sequentially complete topological vector spaces ==

Every [topological vector space](/source/topological_vector_space) is a [uniform space](/source/uniform_space) so the notion of sequential completeness can be applied to them. 

=== Properties of sequentially complete topological vector spaces ===

#A bounded sequentially complete [disk](/source/Absolutely_convex_set) in a Hausdorff topological vector space is a [Banach disk](/source/Banach_disk).{{sfn | Narici|Beckenstein| 2011 | pp=441-442}}
#A Hausdorff locally convex space that is sequentially complete and [bornological](/source/Bornological_space) is [ultrabornological](/source/Ultrabornological_space).{{sfn | Narici|Beckenstein| 2011 | p=449}}

== Examples and sufficient conditions ==

#Every [complete space](/source/complete_space) is sequentially complete but not conversely.
#For metrizable spaces, sequential completeness implies completeness. Together with the previous property, this means sequential completeness and completeness are equivalent over metrizable spaces.
#Every [complete topological vector space](/source/complete_topological_vector_space) is [quasi-complete](/source/Quasi-complete_space) and every quasi-complete topological vector space is sequentially complete.{{sfn | Narici|Beckenstein| 2011 | pp=155-176}}

== See also ==

* [Cauchy net](/source/Cauchy_net)
* [Complete space](/source/Complete_space)
* [Complete topological vector space](/source/Complete_topological_vector_space)
* [Quasi-complete space](/source/Quasi-complete_space)
* [Topological vector space](/source/Topological_vector_space)
* [Uniform space](/source/Uniform_space)

== References ==
{{reflist}}

==Bibliography==
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn | Khaleelulla | {{{year| 1982 }}} | p=}} --> 
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} -->

{{Functional analysis}}
{{TopologicalVectorSpaces}}

<!--- Categories --->

Category:Functional analysis
Category:Topological vector spaces

---
Adapted from the Wikipedia article [Sequentially complete](https://en.wikipedia.org/wiki/Sequentially_complete) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Sequentially_complete?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.