Ultrabarrelled space explained

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)} ^{inf ty} ^{i =1} 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)} ^{inf ty} ^{i =1} 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.

Bibliography

• Bourbaki. Nicolas. Nicolas Bourbaki. Annales de l'Institut Fourier. French. 0042609. 5–16 (1951). Sur certains espaces vectoriels topologiques. 2. 1950. 10.5802/aif.16. free.
  • • • • Book: Robertson. Alex P.. Wendy J.. Robertson . Topological vector spaces. Cambridge Tracts in Mathematics. 53. 1964. Cambridge University Press. 65–75.