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 ${\left(B_i\right)}_{i=1}^{\mathrm{\infty }}$ 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, ${\left(B_i\right)}_{i=1}^{\mathrm{\infty }}$ 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.Template:Sfn

Properties

A locally convex quasi-ultrabarrelled space is quasi-barrelled.Template:Sfn

Examples and sufficient conditions

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

See also

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

References

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• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels