Sequentially complete

In mathematics, specifically in topology and functional analysis, a subspace S of a uniform space X is said to be sequentially complete or semi-complete if every Cauchy sequence in S converges to an element in S. We call X sequentially complete if it is a sequentially complete subset of itself.

Sequentially complete topological vector spaces

Every topological vector space (TVS) is a uniform space so the notion of sequential completeness can be applied to them.

Properties of sequentially complete TVSs
  1. A bounded sequentially complete disk in a Hausdorff TVS is a Banach disk.[1]
  2. A Hausdorff locally convex space that is sequentially complete and bornological is ultrabornological.[2]

Examples and sufficient conditions
  1. Every complete space is sequentially complete but not conversely.
  2. A metrizable space then it is complete if and only if it is sequentially complete.
  3. Every complete topological vector space is quasi-complete and every quasi-complete TVS is sequentially complete.[3]

See also

References
  1. Narici & Beckenstein 2011, pp. 441-442.
  2. Narici & Beckenstein 2011, p. 449.
  3. Narici & Beckenstein 2011, pp. 155-176.

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.
  • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
  • 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.
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