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 ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ 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, ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ 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.[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#Generalisations

References

1. Khaleelulla 1982, pp. 65-76.

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• Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 65–75.
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• 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.
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