Quasi-complete space
In functional analysis, a topological vector space 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 is a normed space and is a quasi-complete locally convex TVS then the set of all compact linear maps of into is a closed vector subspace of.
- Every quasi-complete infrabarrelled space is barreled.
- If 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 has the Heine–Borel property.
Examples and sufficient conditions
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.