Sequentially complete


In mathematics, specifically in topology and functional analysis, a subspace of a uniform space is said to be sequentially complete or semi-complete if every Cauchy sequence in converges to an element in.
We call 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 TVSs


  1. A bounded sequentially complete disk in a Hausdorff TVS 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 TVS is quasi-complete and every quasi-complete TVS is sequentially complete.