Sequential Properties in the Class \(\mathfrak{G}\)

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


In this chapter, we prove that an lcs in the class \(\mathfrak{G}\) is metrizable if and only if E is b-Baire-like if and only if E is Fréchet–Urysohn. Consequently, no proper (LB)-space is Fréchet–Urysohn. We prove that if a (DF)- or (LM)-space E is sequential, then E is either metrizable or Montel (DF). We distinguish a variant of the property C 3 (due to Webb), called property \(C_{3}^{-}\) (i.e., sequential closure of any vector subspace is sequentially closed), and characterize both (DF)-spaces and (LF)-spaces with the property \(C_{3}^{-}\) as being of the form M, ϕ, or M×ϕ, where M is metrizable.


Converse Implication Metrizable Space Sequential Property Sequential Closure Convex Topology 
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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Jerzy Kąkol
    • 1
    Email author
  • 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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