Quasi-complete space
From HandWiki
Short description: A topological vector space in which every closed and bounded subset is complete
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 [math]\displaystyle{ L_b(X;Y) }[/math].[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
References
- ↑ Wilansky 2013, p. 73.
- ↑ 2.0 2.1 2.2 2.3 2.4 Schaefer & Wolff 1999, p. 27.
- ↑ Schaefer & Wolff 1999, p. 201.
- ↑ Schaefer & Wolff 1999, p. 110.
- ↑ 5.0 5.1 Schaefer & Wolff 1999, p. 142.
- ↑ Trèves 2006, p. 520.
- ↑ Narici & Beckenstein 2011, pp. 156-175.
- ↑ Schaefer & Wolff 1999, p. 52.
- ↑ Schaefer & Wolff 1999, p. 144.
- ↑ Khaleelulla 1982, pp. 28-63.
Bibliography
- Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Trèves, François (August 6, 2006). 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.
- Template:Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products
 |  |
