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[edit]

A subset of a TVS is called an ultrabarrel if it is a closed and balanced subset of and if there exists a sequence of closed balanced and absorbing subsets of such that for all In this case, is called a defining sequence for A TVS is called ultrabarrelled if every ultrabarrel in is a neighbourhood of the origin.^[1]

Properties[edit]

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

Examples and sufficient conditions[edit]

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

Counter-examples[edit]

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[edit]

• 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[edit]

Bibliography[edit]

• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). doi:10.5802/aif.16. MR 0042609.
• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. pp. 65–75.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.