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k-spaces and duals of non-archimedean metrizable locally convex spaces

  • Maria Cristina Perez-Garcia EMAIL logo
Published/Copyright: July 4, 2018
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 5

Abstract

The main purpose of this paper is to investigate the non-archimedean counterpart of the classical result stating that the dual of a real or complex metrizable locally convex space, equipped with the locally convex topology of uniform convergence on compact sets, belongs to the topological category formed by the k-spaces. We prove that this counterpart holds when the non-archimedean valued base field 𝕂 is locally compact, but fails for any non-locally compact 𝕂. Here we deal with a topological subcategory, the one formed by the k0-spaces, the adequate non-archimedean substitutes for k-spaces. As a product, we complete some of the achievements on the non-archimedean Banach–Dieudonné Theorem presented in [C. Perez-Garcia and W. H. Schikhof, The p-adic Banach–Dieudonné theorem and semi-compact inductive limits, p-adic Functional Analysis (Poznań 1998), Lecture Notes Pure Appl. Math. 207, Dekker, New York 1999, 295–307]. Also, we use our results to construct in a simple way natural examples of k-spaces (which are also k0-spaces) whose products are not k0-spaces. This in turn improves the, rather involved, example given in [C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over non-Archimedean Valued Fields, Cambridge Stud. Adv. Math. 119, Cambridge University Press, Cambridge, 2010] of two k0-spaces whose product is not a k0-space. Our theory covers an important class of non-archimedean Fréchet spaces, the Köthe sequence spaces, which have a relevant influence on applications such as the definition of a non-archimedean Laplace and Fourier transform.

Keywords: non-archimedean metrizable locally convex spaces; non-archimedean Banach–Dieudonné theorem
MSC 2010: 46S10; 54D50

Communicated by Jörg Brüdern

Funding source: Ministerio de EconomĂ­a y Competitividad

Award Identifier / Grant number: MTM2013-45643-C2-2-P

Funding statement: Research partially supported by Ministerio de EconomĂ­a y Competitividad, Grant MTM2013-45643-C2-2-P.

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Received: 2016-03-22
Revised: 2016-11-03
Published Online: 2018-07-04
Published in Print: 2018-09-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Upper bounds for geodesic periods over rank one locally symmetric spaces
  3. Additive bases with coefficients of newforms
  4. A note on split extensions of bialgebras
  5. Petersson norm of cusp forms associated to real quadratic fields
  6. A realization theorem for sets of lengths in numerical monoids
  7. On the non-existence of the Mackey topology for locally quasi-convex groups
  8. The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces
  9. The cup product of Brooks quasimorphisms
  10. Heat kernel estimates for time fractional equations
  11. Extreme non-Arens regularity of the group algebra
  12. Rational homology and homotopy of high-dimensional string links
  13. Global integrability for solutions to some anisotropic problem with nonstandard growth
  14. Hardy operators on Musielak–Orlicz spaces
  15. Contractibility of the stability manifold for silting-discrete algebras
  16. Order of the canonical vector bundle over configuration spaces of spheres
  17. An endpoint version of uniform Sobolev inequalities
  18. Space-time L2 estimates, regularity and almost global existence for elastic waves
  19. k-spaces and duals of non-archimedean metrizable locally convex spaces
  20. Path homology theory of multigraphs and quivers
https://doi.org/10.1515/forum-2016-0080
Keywords for this article
non-archimedean metrizable locally convex spaces; non-archimedean Banach–Dieudonné theorem

Articles in the same Issue

  1. Frontmatter
  2. Upper bounds for geodesic periods over rank one locally symmetric spaces
  3. Additive bases with coefficients of newforms
  4. A note on split extensions of bialgebras
  5. Petersson norm of cusp forms associated to real quadratic fields
  6. A realization theorem for sets of lengths in numerical monoids
  7. On the non-existence of the Mackey topology for locally quasi-convex groups
  8. The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces
  9. The cup product of Brooks quasimorphisms
  10. Heat kernel estimates for time fractional equations
  11. Extreme non-Arens regularity of the group algebra
  12. Rational homology and homotopy of high-dimensional string links
  13. Global integrability for solutions to some anisotropic problem with nonstandard growth
  14. Hardy operators on Musielak–Orlicz spaces
  15. Contractibility of the stability manifold for silting-discrete algebras
  16. Order of the canonical vector bundle over configuration spaces of spheres
  17. An endpoint version of uniform Sobolev inequalities
  18. Space-time L2 estimates, regularity and almost global existence for elastic waves
  19. k-spaces and duals of non-archimedean metrizable locally convex spaces
  20. Path homology theory of multigraphs and quivers
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