# Sequentially complete

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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. We call X sequentially complete if it is a sequentially complete subset of itself.

## Sequentially complete topological vector spaces

Every topological vector space (TVS) 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 topological vector space is quasi-complete and every quasi-complete TVS is sequentially complete.