Quasi-complete space explained

In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete. This concept is of considerable importance for non-metrizable TVSs.

Properties

• 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.