Quasi-ultrabarrelled space explained

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

\left( B _{i} \right) ^{inf ty} _{i =1} of closed balanced and bornivorous 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) ^{inf ty} _{i =1} 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 . Nicolas Bourbaki . . French . 0042609 . 5–16 (1951) . Sur certains espaces vectoriels topologiques . 2 . 1950. 10.5802/aif.16 . free .
• Book: Robertson . Alex P. . Robertson . Wendy J. . Topological vector spaces . Cambridge Tracts in Mathematics . 53 . 1964 . . 65–75.
• Book: Husain, Taqdir . Barrelledness in topological and ordered vector spaces . Springer-Verlag . Berlin New York . 1978 . 3-540-09096-7 . 4493665 .
• Book: Jarhow, Hans . Locally convex spaces . 1981 . . 978-3-322-90561-1 .