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.

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

 * 1) A bounded sequentially complete disk in a Hausdorff topological vector space is a Banach disk.
 * 2) A Hausdorff locally convex space that is sequentially complete and bornological is ultrabornological.

Examples and sufficient conditions

 * 1) Every complete space is sequentially complete but not conversely.
 * 2) A metrizable space then it is complete if and only if it is sequentially complete.
 * 3) Every complete topological vector space is quasi-complete and every quasi-complete topological vector space is sequentially complete.