Quasi-complete space

Properties

• Every quasi-complete TVS is sequentially complete.
• In a quasi-complete locally convex TVS, the closure of the convex hull of a compact subset is again compact.
• In a quasi-complete Hausdorff TVS, every precompact subset is relatively compact.
• If X is a normed space and Y is a quasi-complete locally convex TVS then the set of all compact linear maps of X into Y is a closed vector subspace of.
• Every quasi-complete infrabarreled space is barreled.
• If X is a quasi-complete locally convex space then every weakly bounded subset of the continuous dual space is strongly bounded.
• A quasi-complete nuclear space then X has the Heine–Borel property.
• In a locally convex quasi-complete space, the closed convex hull of a compact subset is again compact.

  Examples and sufficient conditions

• Every complete TVS is quasi-complete.
• The product of any collection of quasi-complete spaces is again quasi-complete.
• The projective limit of any collection of quasi-complete spaces is again quasi-complete.
• Every semi-reflexive space is quasi-complete.
  
• The quotient of a quasi-complete space by a closed vector subspace may fail to be quasi-complete.

Counter-examples

• There exists an LB-space that is not quasicomplete.