Quasi-complete space

In functional analysis, a topological vector space (TVS) is said to be quasi-complete if every closed and bounded subset is complete.[1] This concept is of considerable importance for non-metrizable TVSs.[1]

Properties
  • Every quasi-complete TVS is sequentially complete.[1]
  • 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.[1]
  • 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 .[2]
  • Every quasi-complete infrabarreled space is barreled.[3]
  • If X is a quasi-complete locally convex space then every weakly bounded subset of the continuous dual space is strongly bounded.[3]
  • A quasi-complete nuclear space then X has the Heine–Borel property.[4]
  • In a locally convex quasi-complete space, the closed convex hull of a compact subset is again compact.[5]

Examples
  • Every complete TVS is quasi-complete.[6]
  • The product of any collection of quasi-complete spaces is again quasi-complete.[1]
  • The projective limit of any collection of quasi-complete spaces is again quasi-complete.[7]
  • Every semi-reflexive space is quasi-complete.[8]
  • 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 quasicomplete.[9]

See also
  • Complete space

References
  1. Schaefer 1999, p. 27.
  2. Schaefer 1999, p. 110.
  3. Schaefer 1999, p. 142.
  4. Treves 2006, p. 520.
  5. Schaefer 1999, p. 201.
  6. Narici 2011, pp. 156-175.
  7. Schaefer 1999, p. 52.
  8. Schaefer 1999, p. 144.
  9. Khaleelulla 1982, pp. 28-63.

Sources
  • Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link)
  • Narici, Lawrence (2011). Topological vector spaces. Boca Raton, FL: CRC Press. ISBN 1-58488-866-0. OCLC 144216834.
  • Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
  • Treves, François (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (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)
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