sequentially complete
{{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
References
{{reflist}}
Bibliography
- {{Khaleelulla Counterexamples in Topological Vector Spaces}}
- {{Rudin Walter Functional Analysis|edition=2}}
- {{Narici Beckenstein Topological Vector Spaces|edition=2}}
- {{Schaefer Wolff Topological Vector Spaces|edition=2}}
- {{Trèves François Topological vector spaces, distributions and kernels}}
{{Functional analysis}}
{{TopologicalVectorSpaces}}