In mathematics, an LB-space, also written (LB)-space, is a topological vector space that is a locally convex inductive limit of a countable inductive system of Banach spaces. This means that is a direct limit of a direct system in the category of locally convex topological vector spaces and each 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 by is identical to the original topology on [1] Some authors (e.g. Schaefer) define the term "LB-space" to mean "strict LB-space."

Definition

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The topology on   can be described by specifying that an absolutely convex subset   is a neighborhood of   if and only if   is an absolutely convex neighborhood of   in   for every  

Properties

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

Examples

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If   is a locally compact topological space that is countable at infinity (that is, it is equal to a countable union of compact subspaces) then the space   of all continuous, complex-valued functions on   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   with the uniform norm and order the family of compact subsets of   by inclusion.[3]

Final topology on the direct limit of finite-dimensional Euclidean spaces

Let

 

denote the space of finite sequences, where   denotes the space of all real sequences. For every natural number   let   denote the usual Euclidean space endowed with the Euclidean topology and let   denote the canonical inclusion defined by   so that its image is

 

and consequently,

 

Endow the set   with the final topology   induced by the family   of all canonical inclusions. With this topology,   becomes a complete Hausdorff locally convex sequential topological vector space that is not a Fréchet–Urysohn space. The topology   is strictly finer than the subspace topology induced on   by   where   is endowed with its usual product topology. Endow the image   with the final topology induced on it by the bijection   that is, it is endowed with the Euclidean topology transferred to it from   via   This topology on   is equal to the subspace topology induced on it by   A subset   is open (resp. closed) in   if and only if for every   the set   is an open (resp. closed) subset of   The topology   is coherent with family of subspaces   This makes   into an LB-space. Consequently, if   and   is a sequence in   then   in   if and only if there exists some   such that both   and   are contained in   and   in  

Often, for every   the canonical inclusion   is used to identify   with its image   in   explicitly, the elements   and   are identified together. Under this identification,   becomes a direct limit of the direct system   where for every   the map   is the canonical inclusion defined by   where there are   trailing zeros.

Counter-examples

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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

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  • DF-space – class of special local-convex space
  • Direct limit – Special case of colimit in category theory
  • Final topology – Finest topology making some functions continuous
  • F-space – Topological vector space with a complete translation-invariant metric
  • LF-space – Topological vector space

Citations

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  1. ^ Schaefer & Wolff 1999, pp. 55–61.
  2. ^ a b c Schaefer & Wolff 1999, pp. 60–63.
  3. ^ a b Schaefer & Wolff 1999, pp. 57–58.
  4. ^ a b Khaleelulla 1982, pp. 28–63.

References

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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.
  • Bierstedt, Klaus-Dieter (1988). "An Introduction to Locally Convex Inductive Limits". Functional Analysis and Applications. Singapore-New Jersey-Hong Kong: Universitätsbibliothek: 35–133. Retrieved 20 September 2020.
  • 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.
  • 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 (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). 16. Providence: American Mathematical Society. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
  • Horváth, John (1966). Topological Vector Spaces and Distributions. Addison-Wesley series in mathematics. Vol. 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. 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.
  • Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 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. 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.