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.

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

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

Complete and metrizable TVSs are ultrabarrelled.^[1] If ${\displaystyle X}$ is a complete locally bounded non-locally convex TVS and if ${\displaystyle B_{0}}$ is a closed balanced and bounded neighborhood of the origin, then ${\displaystyle B_{0}}$ is an ultrabarrel that is not convex and has a defining sequence consisting of non-convex sets.^[1]

There exist barrelled spaces that are not ultrabarrelled.^[1] There exist TVSs that are complete and metrizable (and thus ultrabarrelled) but not barrelled.^[1]

• 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

• 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.