quasi-complete space

• Every quasi-complete TVS is sequentially complete.{{sfn | Schaefer | Wolff | 1999 | p=27}}
• In a quasi-complete locally convex space, the closure of the convex hull of a compact subset is again compact.{{sfn | Schaefer |Wolff| 1999 | p=201}}
• In a quasi-complete Hausdorff TVS, every precompact subset is relatively compact.{{sfn | Schaefer | Wolff | 1999 | p=27}}
• If {{mvar|X}} is a normed space and {{mvar|Y}} is a quasi-complete locally convex TVS then the set of all compact linear maps of {{mvar|X}} into {{mvar|Y}} is a closed vector subspace of $L\_b(X;Y)$.{{sfn | Schaefer | Wolff | 1999 | p=110}}
• Every quasi-complete infrabarrelled space is barreled.{{sfn | Schaefer | Wolff | 1999 | p=142}}
• If {{mvar|X}} is a quasi-complete locally convex space then every weakly bounded subset of the continuous dual space is strongly bounded.{{sfn | Schaefer | Wolff | 1999 | p=142}}
• A quasi-complete nuclear space then {{mvar|X}} has the Heine–Borel property.{{sfn | Trèves | 2006 | p=520}}

Every complete TVS is quasi-complete.{{sfn | Narici | Beckenstein | 2011 | pp=156-175}}

The product of any collection of quasi-complete spaces is again quasi-complete.{{sfn | Schaefer | Wolff | 1999 | p=27}}

The projective limit of any collection of quasi-complete spaces is again quasi-complete.{{sfn | Schaefer | Wolff | 1999 | p=52}}

Every semi-reflexive space is quasi-complete.{{sfn | Schaefer | Wolff | 1999 | p=144}}

The quotient of a quasi-complete space by a closed vector subspace may fail to be quasi-complete.

There exists an LB-space that is not quasi-complete.{{sfn | Khaleelulla | 1982 | pp=28-63}}

• {{annotated link|Complete topological vector space}}
• {{annotated link|Complete uniform space}}

{{reflist|group=note}}

{{reflist|30em}}

• {{Khaleelulla Counterexamples in Topological Vector Spaces}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}
• {{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}}

{{Topological vector spaces}}

Category:Functional analysis