Sequentially complete
In mathematics, specifically in topology and functional analysis, a subspace S of a uniform space X is said to be sequentially complete or semi-complete if every Cauchy sequence in S converges to an element in S. X is called sequentially complete if it is a sequentially complete subset of itself.
01Sequentially 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.
- A Hausdorff locally convex space that is sequentially complete and bornological is ultrabornological.
02Examples 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.
Sources and credits
This article is adapted from the Wikipedia article “Sequentially complete”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
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