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

Properties

A locally convex ultrabarrelled space is a barrelled space. ^[1] Every ultrabarrelled space is a quasi-ultrabarrelled space. ^[1]

Examples and sufficient conditions

Complete and metrizable TVSs are ultrabarrelled. ^[1] If ${\displaystyle X}$ is a complete locally bounded non-locally convex TVS and if ${\displaystyle B_{0}}$ is a closed balanced and bounded neighborhood of the origin, then ${\displaystyle B_{0}}$ is an ultrabarrel that is not convex and has a defining sequence consisting of non-convex sets. ^[1]

Counter-examples

There exist barrelled spaces that are not ultrabarrelled. ^[1] There exist TVSs that are complete and metrizable (and thus ultrabarrelled) but not barrelled. ^[1]

See also

• Barrelled space  – Type of topological vector space
• Countably barrelled space
• Countably quasi-barrelled space
• Infrabarreled space
• Uniform boundedness principle#Generalisations  – A theorem stating that pointwise boundedness implies uniform boundedness

Citations

1. Khaleelulla 1982, pp. 65–76.

Bibliography

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