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 B₀ 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 ⊆ B_i for all i = 0, 1, .... In this case, $\left(B_{i}\right)_{i=1}^{\infty }$ is called a defining sequence for B₀. A TVS X is called ultrabarrelled if every ultrabarrel in X is a neighbourhood of the origin.

Properties

A locally convex ultrabarrelled space is barrelled. 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 is a closed balanced and bounded neighborhood of the origin, then B 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
• Countably barrelled space
• Countably quasi-barrelled space
• Infrabarreled space
• Uniform boundedness principle#Generalisations

References

• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
• Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. pp. 65–75.
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665. {{cite book}}: Invalid |ref=harv (help)
• Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1. {{cite book}}: Invalid |ref=harv (help)
• Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. Vol. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. {{cite book}}: Invalid |ref=harv (help)
• Schaefer, Helmut H. (1971). Topological vector spaces. GTM. Vol. 3. New York: Springer-Verlag. p. 60. ISBN 0-387-98726-6.
• Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. {{cite book}}: Invalid |ref=harv (help)
• Treves, Francois (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. {{cite book}}: Invalid |ref=harv (help)