Quasi-complete space

• Every quasi-complete TVS is sequentially complete. • In a quasi-complete locally convex space, 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 is a normed space and is a quasi-complete locally convex TVS then the set of all compact linear maps of into is a closed vector subspace of L_b(X;Y). • Every quasi-complete infrabarrelled space is barreled. • If 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 has the Heine–Borel property. == 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 quasi-complete. == See also ==