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 $\left(B_{i}\right)_{i=1}^{\infty }$ of closed balanced and bornivorous subsets of X such that B_i+1 + B_i+1 ⊆ B_i for all i = 0, 1, .... In this case, $\left(B_{i}\right)_{i=1}^{\infty }$ 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.

Properties

A locally convex quasi-ultrabarrelled space is quasi-barrelled.

Examples and sufficient conditions

Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled. Complete and metrizable TVSs are quasi-ultrabarrelled.

See also

• Barrelled space
• Countably barrelled space
• Countably quasi-barrelled space
• Infrabarreled space
• Ultrabarrelled space
• Uniform boundedness principle#Generalisations

References

• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). doi:10.5802/aif.16. MR 0042609.
• Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 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.
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• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 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.