Quasi-complete space

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

Properties

• Every quasi-complete TVS is sequentially complete.Template:Sfn
• In a quasi-complete locally convex space, the closure of the convex hull of a compact subset is again compact.Template:Sfn
• In a quasi-complete Hausdorff TVS, every precompact subset is relatively compact.Template:Sfn
• If Template:Mvar is a normed space and Template:Mvar is a quasi-complete locally convex TVS then the set of all compact linear maps of Template:Mvar into Template:Mvar is a closed vector subspace of $L_b(X;Y)$.Template:Sfn
• Every quasi-complete infrabarrelled space is barreled.Template:Sfn
• If Template:Mvar is a quasi-complete locally convex space then every weakly bounded subset of the continuous dual space is strongly bounded.Template:Sfn
• A quasi-complete nuclear space then Template:Mvar has the Heine–Borel property.Template:Sfn

Examples and sufficient conditions

Every complete TVS is quasi-complete.Template:Sfn The product of any collection of quasi-complete spaces is again quasi-complete.Template:Sfn The projective limit of any collection of quasi-complete spaces is again quasi-complete.Template:Sfn Every semi-reflexive space is quasi-complete.Template:Sfn

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.Template:Sfn

See also

• Template:Annotated link
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Bibliography

• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels
• Template:Wilansky Modern Methods in Topological Vector Spaces
• Template:Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products