A revised closed graph theorem for quasi-Suslin spaces



Some results about the continuity of special linear maps between F-spaces recently obtained by Drewnowski have motivated us to revise a closed graph theorem for quasi-Suslin spaces due to Valdivia. We extend Valdivia’s theorem by showing that a linear map with closed graph from a Baire tvs into a tvs admitting a relatively countably compact resolution is continuous. This also applies to extend a result of De Wilde and Sunyach. A topological space X is said to have a (relatively countably) compact resolution if X admits a covering {Aα:α ∈ ℕ} consisting of (relatively countably) compact sets such that AαAβ for αβ. Some applications and two open questions are provided.


K-analytic space web space quasi-Suslin space 

MSC 2000

54C14 54D08 46A03 


  1. [1]
    B. Cascales: On K-analytic locally convex spaces. Arch. Math. 49 (1987), 232–244.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    B. Cascales, J. Orihuela: On compactness in locally convex spaces. Math. Z. 195 (1987), 365–381.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J.P.R. Christensen: Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, Vol. 10. North Holland, Amsterdam, 1974.MATHGoogle Scholar
  4. [4]
    W.W. Comfort, D. Remus: Compact groups of Ulam-measurable cardinality: Partial converse theorems of Arkhangel’skii and Varopoulos. Math. Jap. 39 (1994), 203–210.MATHMathSciNetGoogle Scholar
  5. [5]
    P. Dierolf, S. Dierolf, L. Drewnowski: Remarks and examples concerning unordered Baire-like and ultrabarrelled spaces. Colloq. Math. 39 (1978), 109–116.MATHMathSciNetGoogle Scholar
  6. [6]
    L. Drewnowski: Resolutions of topological linear spaces and continuity of linear maps. J. Math. Anal. Appl. 335 (2007), 1177–1194.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    L. Drewnowski: The dimension and codimension of analytic subspaces in topological vector spaces, with applications to the constructions of some pathological topological vector spaces. Liège 1982 (unpublished Math. talk).Google Scholar
  8. [8]
    L. Drewnowski, I. Labuda: Sequence F-spaces of L 0-type over submeasures of ℕ. To appear.Google Scholar
  9. [9]
    J. Kakol, M. López Pellicer: Compact coverings for Baire locally convex spaces. J. Math. Anal. Appl. 332 (2007), 965–974.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    J. L. Kelley, I. Namioka et al.: Linear Topological Spaces. Van Nostrand, London, 1963.MATHGoogle Scholar
  11. [11]
    Y. Kōmura: On linear topological spaces. Kumamoto J. Sci., Ser. A 5 (1962), 148–157.MATHMathSciNetGoogle Scholar
  12. [12]
    M. Nakamura: On quasi-Suslin space and closed graph theorem. Proc. Japan Acad. 46(1970), 514–517.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Nakamura: On closed graph theorem. Proc. Japan Acad. 46 (1970), 846–849.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    P. Perez Carreras, J. Bonet: Barrelled Locally Convex Spaces, Vol. 131. North Holland, Amsterdam, 1987.Google Scholar
  15. [15]
    C.A. Rogers, J. E. Jayne, C. Dellacherie, F. Topsoe, J. Hoffman-Jorgensen, D.A. Martin, A. S. Kechris, A.H. Stone: Analytic Sets. Academic Press, London, 1980.MATHGoogle Scholar
  16. [16]
    M. Talagrand: Espaces de Banach faiblement K-analytiques. Ann. Math. 110 (1979), 407–438.CrossRefMathSciNetGoogle Scholar
  17. [17]
    V.V. Tkachuk: A space C p(X) is dominated by irrationals if and only if it is K-analytic. Acta Math. Hungar. 107 (2005), 253–265.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    M. Valdivia: Topics in Locally Convex Spaces. North-Holland, Amsterdam, 1982.MATHGoogle Scholar
  19. [19]
    M. Valdivia: Quasi-LB-spaces. J. Lond. Math. Soc. 35 (1987), 149–168.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2009

Authors and Affiliations

  1. 1.Centro de Investigación OperativaUniversidad Miguel HernándezElche (Alicante)Spain
  2. 2.Faculty of Mathematics and InformaticsA. Mickiewicz UniversityPoznańPoland
  3. 3.Depto. de Matemática Aplicada and IMPAUniversidad Politécnica de ValenciaValenciaSpain

Personalised recommendations