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In mathematics, specifically in [[Topology|topology]] and [[Functional analysis|functional analysis]], a subspace {{mvar|S}} of a [[Uniform space|uniform space]] {{mvar|X}} is said to be '''sequentially complete''' or '''semi-complete''' if every [[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|topological vector space]] is a [[Uniform space|uniform space]] so the notion of sequential completeness can be applied to them. 

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

#A bounded sequentially complete [[Absolutely convex set|disk]] in a Hausdorff topological vector space is a Banach disk.{{sfn | Narici|Beckenstein| 2011 | pp=441-442}}
#A Hausdorff locally convex space that is sequentially complete and [[Bornological space|bornological]] is [[Ultrabornological space|ultrabornological]].{{sfn | Narici|Beckenstein| 2011 | p=449}}

== Examples and sufficient conditions ==

#Every complete space is sequentially complete but not conversely.
#A metrizable space then it is complete if and only if it is sequentially complete.
#Every [[Complete topological vector space|complete topological vector space]] is [[Quasi-complete space|quasi-complete]] and every quasi-complete topological vector space is sequentially complete.{{sfn | Narici|Beckenstein| 2011 | pp=155-176}}

== See also ==

* Cauchy net
* Complete space
* [[Complete topological vector space]]
* [[Quasi-complete space]]
* [[Topological vector space]]
* [[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]]

{{Sourceattribution|Sequentially complete}}