In functional analysis and related areas of mathematics, a quasi-ultrabarrelled space is a topological vector spaces (TVS) for which every bornivorous ultrabarrel is a neighbourhood of the origin.
Definition A subset B0 of a TVS X is called a bornivorous ultrabarrel if it is a closed, balanced, and bornivorous subset of X and if there exists a sequence ( B i ) i = 1 ∞ {\displaystyle \left(B_{i}\right)_{i=1}^{\infty }} of closed balanced and bornivorous subsets of X such that Bi+1 + Bi+1 ⊆ Bi for all i = 0, 1, .... In this case, ( B i ) i = 1 ∞ {\displaystyle \left(B_{i}\right)_{i=1}^{\infty }} is called a defining sequence for B0. A TVS X is called quasi-ultrabarrelled if every bornivorous ultrabarrel in X is a neighbourhood of the origin.
Properties A locally convex quasi-ultrabarrelled space is quasi-barrelled.
Examples and sufficient conditions Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled. Complete and metrizable TVSs are quasi-ultrabarrelled.
See also Barrelled space Countably barrelled space Countably quasi-barrelled space Infrabarreled space Ultrabarrelled 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). doi:10.5802/aif.16. 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. Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1. 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. 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.
