Quasi-complete space

In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete^[1] if every closed and bounded subset is complete.^[2] This concept is of considerable importance for non-metrizable TVSs.^[2]

Properties

• Every quasi-complete TVS is sequentially complete.^[2]
• In a quasi-complete locally convex space, the closure of the convex hull of a compact subset is again compact.^[3]
• In a quasi-complete Hausdorff TVS, every precompact subset is relatively compact.^[2]
• 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 ${\displaystyle L_{b}(X;Y)}$.^[4]
• Every quasi-complete infrabarrelled space is barreled.^[5]
• If X is a quasi-complete locally convex space then every weakly bounded subset of the continuous dual space is strongly bounded.^[5]
• A quasi-complete nuclear space then X has the Heine–Borel property.^[6]

Examples and sufficient conditions

Every complete TVS is quasi-complete.^[7] The product of any collection of quasi-complete spaces is again quasi-complete.^[2] The projective limit of any collection of quasi-complete spaces is again quasi-complete.^[8] Every semi-reflexive space is quasi-complete.^[9]

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.^[10]

See also

• Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
• Complete uniform space – Topological space with a notion of uniform propertiesPages displaying short descriptions of redirect targets