Quasi-complete space
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 TVS, 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.[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 .[3]
- Every quasi-complete infrabarreled space is barreled.[4]
- If X is a quasi-complete locally convex space then every weakly bounded subset of the continuous dual space is strongly bounded.[4]
- A quasi-complete nuclear space then X has the Heine–Borel property.[5]
- In a locally convex quasi-complete space, the closed convex hull of a compact subset is again compact.[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.
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See also
- Complete space
References
- Wilansky 2013, p. 73.
- Schaefer 1999, p. 27.
- Schaefer 1999, p. 110.
- Schaefer 1999, p. 142.
- Treves 2006, p. 520.
- Schaefer 1999, p. 201.
- Narici 2011, pp. 156-175.
- Schaefer 1999, p. 52.
- Schaefer 1999, p. 144.
- Khaleelulla 1982, pp. 28-63.
- 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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Narici, Lawrence (2011). Topological vector spaces. Boca Raton, FL: CRC Press. ISBN 1-58488-866-0. 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.CS1 maint: ref=harv (link)
- Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
- Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.CS1 maint: ref=harv (link)
- Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.CS1 maint: ref=harv (link)
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