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 $B_0$ of a TVS $X$ is called an ultrabarrel if it is a closed and balanced subset of $X$ and if there exists a sequence ${\left(B_i\right)}_{i=1}^{\mathrm{\infty }}$ of closed balanced and absorbing subsets of $X$ such that $B_{i+1}+B_{i+1}\subseteq B_i$ for all $i=0,1,\dots .$ In this case, ${\left(B_i\right)}_{i=1}^{\mathrm{\infty }}$ is called a defining sequence for $B_0.$ A TVS $X$ is called ultrabarrelled if every ultrabarrel in $X$ is a neighbourhood of the origin.Template:Sfn

Properties

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

Examples and sufficient conditions

Complete and metrizable TVSs are ultrabarrelled.Template:Sfn If $X$ is a complete locally bounded non-locally convex TVS and if $B_0$ is a closed balanced and bounded neighborhood of the origin, then $B_0$ is an ultrabarrel that is not convex and has a defining sequence consisting of non-convex sets.Template:Sfn

Counter-examples

There exist barrelled spaces that are not ultrabarrelled.Template:Sfn There exist TVSs that are complete and metrizable (and thus ultrabarrelled) but not barrelled.Template:Sfn

See also

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Bibliography

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• Template:Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces
• Template:Jarchow Locally Convex Spaces
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Cite book
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels