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 [math]\displaystyle{ B_0 }[/math] of a TVS [math]\displaystyle{ X }[/math] is called an ultrabarrel if it is a closed and balanced subset of [math]\displaystyle{ X }[/math] and if there exists a sequence [math]\displaystyle{ \left(B_i\right)_{i=1}^{\infty} }[/math] of closed balanced and absorbing subsets of [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ B_{i+1} + B_{i+1} \subseteq B_i }[/math] for all [math]\displaystyle{ i = 0, 1, \ldots. }[/math] In this case, [math]\displaystyle{ \left(B_i\right)_{i=1}^{\infty} }[/math] is called a defining sequence for [math]\displaystyle{ B_0. }[/math] A TVS [math]\displaystyle{ X }[/math] is called ultrabarrelled if every ultrabarrel in [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X }[/math] is a complete locally bounded non-locally convex TVS and if [math]\displaystyle{ B_0 }[/math] is a closed balanced and bounded neighborhood of the origin, then [math]\displaystyle{ B_0 }[/math] 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

Bibliography

• {{cite journal
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