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.

## Contents

* 1 Definition
* 2 Properties
* 3 Characterizations
* 4 Examples and sufficient conditions
* 4.1 Counter-examples
* 5 See also
* 6 References
* 7 Bibliography

## 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

1. ↑ Jarchow 1981, p. 222.
2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Khaleelulla 1982, pp. 28-63.
3. ↑ Khaleelulla 1982, p. 28.
4. ↑ Khaleelulla 1982, pp. 35.
5. ↑ Adasch, Ernst & Keim 1978, p. 43.
6. ↑ Adasch, Ernst & Keim 1978, pp. 70-73.
7. ↑ Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)

## 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.

* v
* t
* e

Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
* Closed
* Compact (on Hilbert spaces)
* (Dis)Continuous
* Densely defined
* Fredholm
* Hilbert–Schmidt
* Functionals (positive)
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

* Banach algebras
* C*-algebras
* Spectrum (C*-algebra
* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
* Banach–Mazur
* Banach–Saks
* Banach–Schauder (open mapping)
* Banach–Steinhaus (Uniform boundedness)
* Bessel's inequality
* Cauchy–Schwarz inequality
* Closed graph
* Closed range
* Eberlein–Šmulian
* Freudenthal spectral
* Gelfand–Mazur
* Gelfand–Naimark
* Goldstine
* Hahn–Banach (hyperplane separation)
* Kakutani fixed-point
* Krein–Milman
* Lomonosov's invariant subspace
* Mackey–Arens
* Mazur's lemma
* M. Riesz extension
* Riesz representation
* Parseval's identity
* Schauder fixed-point

Analysis|

* Abstract Wiener space
* Bochner space
* Differentiation in Fréchet spaces
* Derivatives (Fréchet
* Gateaux
* functional
* holomorphic)
* Integrals (Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak)
* Functional calculus (Borel
* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
* Projection-valued
* Vector)
* Weakly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Balanced
* Bounded
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Minkowski addition
* Polar

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Quasibarrelled space. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Quasibarrelled_space&oldid=2235154"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template