Quasi-ultrabarrelled space
In functional analysis and related areas of mathematics, a quasi-ultrabarrelled space is a topological vector spaces (TVS) for which every bornivorous ultrabarrel is a neighbourhood of the origin.
Definition
A subset B0 of a TVS X is called a bornivorous ultrabarrel if it is a closed, balanced, and bornivorous subset of X and if there exists a sequence [math]\displaystyle{ \left( B_{i} \right)_{i=1}^{\infty} }[/math] of closed balanced and bornivorous subsets of X such that Bi+1 + Bi+1 ⊆ Bi for all i = 0, 1, .... In this case, [math]\displaystyle{ \left( B_{i} \right)_{i=1}^{\infty} }[/math] is called a defining sequence for B0. A TVS X is called quasi-ultrabarrelled if every bornivorous ultrabarrel in X is a neighbourhood of the origin.[1]
Properties
A locally convex quasi-ultrabarrelled space is quasi-barrelled.[1]
Examples and sufficient conditions
Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled. Complete and metrizable TVSs are quasi-ultrabarrelled.[1]
See also
- Barrelled space
- Countably barrelled space
- Countably quasi-barrelled space
- Infrabarreled space
- Ultrabarrelled space
- Uniform boundedness principle
References
- ↑ 1.0 1.1 1.2 Khaleelulla 1982, pp. 65-76.
- "Sur certains espaces vectoriels topologiques" (in French). Annales de l'Institut Fourier 2: 5–16 (1951). 1950. doi:10.5802/aif.16. http://www.numdam.org/item?id=AIF_1950__2__5_0.
- 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.
- Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1.
- 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.
- 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 (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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