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

Properties

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

Examples and sufficient conditions

Complete and metrizable TVSs are ultrabarrelled. 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.

Counter-examples

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

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

Bibliography

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