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 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 ${\displaystyle L_{b}(X;Y)}$.[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
• Complete uniform space

References

1. Wilansky 2013, p. 73.
2. Schaefer & Wolff 1999, p. 27.
3. Schaefer & Wolff 1999, p. 201.
4. Schaefer & Wolff 1999, p. 110.
5. Schaefer & Wolff 1999, p. 142.
6. Trèves 2006, p. 520.
7. Narici & Beckenstein 2011, pp. 156-175.
8. Schaefer & Wolff 1999, p. 52.
9. Schaefer & Wolff 1999, p. 144.
10. Khaleelulla 1982, pp. 28-63.

Bibliography

• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 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. 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. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.