LB-space

In mathematics, an LB-space, also written (LB)-space, is a topological vector space X that is a locally convex inductive limit of a countable inductive system of Banach spaces. This means that X is a direct limit of a direct system in the category of locally convex topological vector spaces and each Xn is a Banach space.

If each of the bonding maps is an embedding of TVSs then the LB-space is called a strict LB-space. This means that the topology induced on Xn by Xn+1> is identical to the original topology on Xn.[1] Some authors (e.g. Schaefer) define the term "LB-space" to mean "strict LB-space," so when reading mathematical literature, its recommended to always check how LB-space is defined.

Definition

The topology on X can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if is an absolutely convex neighborhood of 0 in Xn for every n.

Properties

A strict LB-space is complete,[2] barrelled,[2] and bornological[2] (and thus ultrabornological).

Examples

If D is a locally compact topological space that is countable at infinity (i.e. equal to a countable union of compact subspaces) then the space of all continuous, complex-valued functions on D with compact support is a strict LB-space.[3] For any compact subset , let denote the Banach space of complex-valued functions that are supported by K with the uniform norm and order the family of compact subsets of D by inclusion.[3]

Counter-examples

There exists a bornological LB-space whose strong bidual is not bornological.[4] There exists an LB-space that is not quasi-complete.[4]

See also

References
    1. Schaefer & Wolff 1999, pp. 55-61.
    2. Schaefer & Wolff 1999, pp. 60-63.
    3. Schaefer & Wolff 1999, pp. 57-58.
    4. Khaleelulla 1982, pp. 28-63.
    • Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
    • Bierstedt, Klaus-Dieter (1988). An Introduction to Locally Convex Inductive Limits. Functional Analysis and Applications. Singapore-New Jersey-Hong Kong: Universitätsbibliothek. pp. 35–133. MR 0046004. Retrieved 20 September 2020.
    • Bourbaki, Nicolas (1987) [1981]. Sur certains espaces vectoriels topologiques [Topological Vector Spaces: Chapters 1–5]. Annales de l'Institut Fourier. Éléments de mathématique. 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.
    • Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
    • 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.
    • Horváth, John (1966). Topological Vector Spaces and Distributions. Addison-Wesley series in mathematics. 1. Reading, MA: Addison-Wesley Publishing Company. ISBN 978-0201029857.
    • 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. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
    • Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 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. 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.
    • Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
    • 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 (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.
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