Infrabarrelled space

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In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infrabarreled) if every bounded barrel is a neighborhood of the origin.[1]

Similarly, 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

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A subset   of a topological vector space (TVS)   is called bornivorous if it absorbs all bounded subsets of  ; that is, if for each bounded subset   of   there exists some scalar   such that   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.[2][3]

Characterizations

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If   is a Hausdorff locally convex space then the canonical injection from   into its bidual is a topological embedding if and only if   is infrabarrelled.[4]

A Hausdorff topological vector space   is quasibarrelled if and only if every bounded closed linear operator from   into a complete metrizable TVS is continuous.[5] By definition, a linear   operator is called closed if its graph is a closed subset of  

For a locally convex space   with continuous dual   the following are equivalent:

  1.   is quasibarrelled.
  2. Every bounded lower semi-continuous semi-norm on   is continuous.
  3. Every  -bounded subset of the continuous dual space   is equicontinuous.

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

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

Properties

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Every quasi-complete infrabarrelled space is barrelled.[1]

A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled.[6]

A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled.[7]

A locally convex quasibarrelled space that is also a σ-barrelled space is necessarily a barrelled space.[3]

A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled.[3]

Examples

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Every barrelled space is infrabarrelled.[1] A closed vector subspace of an infrabarrelled space is, however, not necessarily infrabarrelled.[8]

Every product and locally convex direct sum of any family of infrabarrelled spaces is infrabarrelled.[8] Every separated quotient of an infrabarrelled space is infrabarrelled.[8]

Every Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled.[9] Thus, every metrizable TVS is quasibarrelled.

Note that there exist quasibarrelled spaces that are neither barrelled nor bornological.[3] There exist Mackey spaces that are not quasibarrelled.[3] There exist distinguished spaces, DF-spaces, and  -barrelled spaces that are not quasibarrelled.[3]

The strong dual space   of a Fréchet space   is distinguished if and only if   is quasibarrelled.[10]

Counter-examples

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There exists a DF-space that is not quasibarrelled.[3]

There exists a quasibarrelled DF-space that is not bornological.[3]

There exists a quasibarrelled space that is not a σ-barrelled space.[3]

See also

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References

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Bibliography

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  • Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
  • Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
  • Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
  • Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
  • Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
  • Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
  • Hogbe-Nlend, Henri (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. MR 0500064. OCLC 316549583.
  • Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 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.
  • 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.
  • Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
  • Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.
  • 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. Vol. 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 (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
  • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
  • Wong, Yau-Chuen (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. Vol. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.