Sequentially complete
# 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

* 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.

## Examples and sufficient conditions

* 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.
