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. Every topological vector space is a uniform space so the notion of sequential completeness can be applied to them. 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. 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 is quasi-complete and every quasi-complete topological vector space is sequentially complete.