Ultrabarrelled space

In functional analysis and related areas of mathematics, an ultrabarrelled space is a topological vector spaces (TVS) for which every ultrabarrel is a neighbourhood of the origin.

Definition

A subset B0 of a TVS X is called an ultrabarrel if it is a closed and balanced subset of X and if there exists a sequence of closed balanced and absorbing subsets of X such that Bi+1 + Bi+1 ⊆ Bi for all i = 0, 1, .... In this case, is called a defining sequence for B0. A TVS X is called ultrabarrelled if every ultrabarrel in X is a neighbourhood of the origin.[1]

Properties

A locally convex ultrabarrelled space is barrelled.[1] Every ultrabarrelled space is a quasi-ultrabarrelled space.[1]

Examples and sufficient conditions

Complete and metrizable TVSs are ultrabarrelled.[1] If X is a complete locally bounded non-locally convex TVS and if B is a closed balanced and bounded neighborhood of the origin, then B is an ultrabarrel that is not convex and has a defining sequence consisting of non-convex sets.[1]

Counter-examples

There exist barrelled spaces that are not ultrabarrelled.[1] There exist TVSs that are complete and metrizable (and thus ultrabarrelled) but not barrelled.[1]

See also

References
  1. Khaleelulla 1982, pp. 65-76.
    • Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
    • Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 65–75.
    • Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.CS1 maint: ref=harv (link)
    • Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1.CS1 maint: ref=harv (link)
    • 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)
    • Schaefer, Helmut H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. p. 60. ISBN 0-387-98726-6.
    • 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, Francois (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)
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