Quasibarrelled Space

In functional analysis and related areas of mathematics, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Quasibarrelled spaces are studied because they are a weakening of the defining condition of barrelled spaces, for which a form of the Banach–Steinhaus theorem holds.

Definition

A subset [math]\displaystyle{ B }[/math] of a topological vector space (TVS) [math]\displaystyle{ X }[/math] is called bornivorous if it absorbs all bounded subsets of [math]\displaystyle{ X }[/math]; that is, if for each bounded subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ X, }[/math] there exists some scalar [math]\displaystyle{ r }[/math] such that [math]\displaystyle{ S \subseteq r B. }[/math] 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]

Properties

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 quasibarrelled space that is also a σ-barrelled space is necessarily a barrelled space.^[2]

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

Characterizations

A Hausdorff topological vector space [math]\displaystyle{ X }[/math] is quasibarrelled if and only if every bounded closed linear operator from [math]\displaystyle{ X }[/math] into a complete metrizable TVS is continuous.^[5] By definition, a linear [math]\displaystyle{ F : X \to Y }[/math] operator is called closed if its graph is a closed subset of [math]\displaystyle{ X \times Y. }[/math]

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

1. [math]\displaystyle{ X }[/math] is quasibarrelled.
2. Every bounded lower semi-continuous semi-norm on [math]\displaystyle{ X }[/math] is continuous.
3. Every [math]\displaystyle{ \beta(X', X) }[/math]-bounded subset of the continuous dual space [math]\displaystyle{ X^{\prime} }[/math] is equicontinuous.

If [math]\displaystyle{ X }[/math] is a metrizable locally convex TVS then the following are equivalent:

1. The strong dual of [math]\displaystyle{ X }[/math] is quasibarrelled.
2. The strong dual of [math]\displaystyle{ X }[/math] is barrelled.
3. The strong dual of [math]\displaystyle{ X }[/math] is bornological.

Examples and sufficient conditions

Every Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled.^[6] Thus, every metrizable TVS is quasibarrelled.

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 [math]\displaystyle{ \sigma }[/math]-barrelled spaces that are not quasibarrelled.^[2]

The strong dual space [math]\displaystyle{ X_b^{\prime} }[/math] of a Fréchet space [math]\displaystyle{ X }[/math] is distinguished if and only if [math]\displaystyle{ X }[/math] is quasibarrelled.^[7]

Counter-examples

There exists a DF-space that is not quasibarrelled.^[2] There exists a quasibarrelled DF-space that is not bornological.^[2] There exists a quasibarrelled space that is not a σ-barrelled space.^[2]

See also

• Barrelled space – Type of topological vector space
• Countably barrelled space
• Infrabarrelled space

References

Bibliography

• Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. {3834. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. 
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. 
• Bourbaki, Nicolas (1987). Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. 2. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190. http://www.numdam.org/item?id=AIF_1950__2__5_0. 
• Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. 
• Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138. 
• Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. https://archive.org/details/topologicalvecto0000grot. 
• Template:Hogbe-Nlend Bornologies and Functional Analysis
• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. 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. 
• Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. OCLC 840293704. 
• Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin 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. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. 

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