{{more footnotes|date=May 2020}}

In mathematics, specifically in topology and functional analysis, a subspace {{mvar|S}} of a 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 is a 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 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 is ultrabornological.{{sfn | Narici|Beckenstein| 2011 | p=449}}

== Examples and sufficient conditions ==

#Every 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 is 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}}

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Category:Functional analysis Category:Topological vector spaces