Quasibarrelled space

A subset B of a TVS X is called bornivorous if it absorbs all bounded subsets of X; that is, if for each bounded subset S of X, there exists some scalar r such that S ⊆ rB. A barrelled set or a barrel in a TVS is a set which is convex, balanced, absorbing and closed. A quasibarrelled space is a TVS for which every bornivorous barrelled set in the space is a neighbourhood of the origin.[1][2]

A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled.[3] A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled.[4]

A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled.[2]

A locally convex quasi-barreled space that is also a 𝜎-barrelled space is a barrelled space.[2]

For a locally convex space X with continuous dual ${\displaystyle X^{\prime }}$ the following are equivalent:

• X is quasi-barrelled,
• every bounded lower semi-continuous semi-norm on X is continuous,
• every ${\displaystyle \beta (X',X)}$-bounded subset of the continuous dual space ${\displaystyle X'}$ is equicontinuous.

If X is a metrizable locally convex TVS then the following are equivalent:

1. The strong dual of X is quasibarrelled.
2. The strong dual of X is barrelled.
3. The strong dual of X is bornological.

Every locally convex Hausdorff barrelled space is quasibarreled.[3] Every locally convex Hausdorff bornological space is quasibarreled.[3] Thus, every locally convex metrizable TVS is quasibarreled.[3]

Note that there exist quasibarrelled spaces that are neither barrelled nor bornological.[2] There exist Mackey spaces that are not quasibarrelled.[2] There exist distinguished spaces, DF-spaces, and ${\displaystyle \sigma }$-barrelled spaces that are not quasibarrelled.[2]

• Barrelled space
• Countably barrelled space
• Countably quasi-barrelled space
• Infrabarreled space
• Uniform boundedness principle#Generalisations

• 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. 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.CS1 maint: ref=harv (link)
• Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1.CS1 maint: ref=harv (link)
• Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link)
• Schaefer, Helmut H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. p. 60. ISBN 0-387-98726-6.
• Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
• Treves, Francois (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link)