Quasi-complete space

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 $X$ 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 $$L_b(X;Y)$$.
 * Every quasi-complete infrabarrelled 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.

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.