Weakly Realcompact Locally Convex Spaces

  • Jerzy Kąkol
  • Wiesław Kubiś
  • Manuel López-Pellicer
Part of the Developments in Mathematics book series (DEVM, volume 24)

Abstract

In this chapter, we continue the study of spaces in the class \(\mathfrak{G}\). We prove that the weak dual (E′,σ(E′,E)) of an lcs E in the class \(\mathfrak{G}\) is K-analytic if and only if (E′,σ(E′,E)) is Lindelöf if and only if (E,σ(E,E′)) has countable tightness. We show that every quasibarrelled space in the class \(\mathfrak{G}\) has countable tightness both for the weak and the original topologies. This extends a classical result of Kaplansky for a metrizable lcs. Although (DF)-spaces belong to the class \(\mathfrak{G}\), concrete examples of (DF)-spaces without countable tightness are provided. On the other hand, there are many Banach spaces E for which E endowed with the weak topology is not Lindelöf. We show, however (following Khurana), that every WCG Fréchet space E is weakly K-analytic. An example due to Pol showing that there exists a Banach space C(X) over a compact scattered space X such that C(X) is weakly Lindelöf and not WCG is presented. We show (after Amir and Lindenstrauss) that every nonseparable reflexive Banach space contains a complemented separable subspace. Several consequences are provided.

Keywords

Hull Radon Corson 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Jerzy Kąkol
    • 1
  • Wiesław Kubiś
    • 2
    • 3
  • Manuel López-Pellicer
    • 4
    • 5
  1. 1.Faculty of Mathematics and InformaticsA. Mickiewicz UniversityPoznanPoland
  2. 2.Institute of MathematicsJan Kochanowski UniversityKielcePoland
  3. 3.Institute of MathematicsAcademy of Sciences of the Czech RepublicPraha 1Czech Republic
  4. 4.IUMPAUniversitat Poltècnica de ValènciaValenciaSpain
  5. 5.Royal Academy of SciencesMadridSpain

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