• Source: Quasi-complete space

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 – 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




References






Bibliography


Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
Wong, Yau-Chuen (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. Vol. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.

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