title: Sequentially complete
text: # 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.
  