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 ) {\displaystyle 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.
See also Complete topological vector space – Structure in functional analysis Complete uniform space – Topological space with a notion of uniform propertiesPages displaying short descriptions of redirect targets
References
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