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Quasi-complete space
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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
[edit]- 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 .[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
[edit]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
[edit]See also
[edit]- Complete topological vector space – Structure in functional analysis
- Complete uniform space – Topological space with a notion of uniform properties
References
[edit]- ^ Wilansky 2013, p. 73.
- ^ a b c d e Schaefer & Wolff 1999, p. 27.
- ^ Schaefer & Wolff 1999, p. 201.
- ^ Schaefer & Wolff 1999, p. 110.
- ^ a b 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
[edit]- 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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