Archiv der Mathematik

, Volume 101, Issue 1, pp 65–77 | Cite as

A quantitative version of Krein’s theorems for Fréchet spaces

  • Carlos Angosto
  • Jerzy Ka̧kol
  • Albert Kubzdela
  • Manuel López-Pellicer


For a Banach space E and its bidual space E′′, the following function \({k(H) : = {\rm sup}_{y\in\overline{H}^{\sigma(E^{\prime \prime},E^{\prime})}} {\rm inf}_{x\in E} \|y - x\|}\) defined on bounded subsets H of E measures how far H is from being σ(E, E′)-relatively compact in E. This concept, introduced independently by Granero [10] and Cascales et al. [7], has been used to study a quantitative version of Krein’s theorem for Banach spaces E and spaces Cp(K) over compact K. In the present paper, a quantitative version of Krein’s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E the above function k(H) reads as follows \({k(H) := {\rm sup}\{d(h, E) : h \in \overline{H}^{\sigma(E^{\prime \prime},E^{\prime})}\},}\) where d(h, E) is the natural distance of h to E in the bidual E′′. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds\({k(coH) < (2^{n+1} - 2) k(H) + \frac{1}{2^{n}}}\)for all\({n \in \mathbb{N}}\). Consequently this yields also the following formula\({k(coH) \leq \sqrt{k(H)}(3 - 2\sqrt{k(H)})}\). Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein’s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet space. We also define and discuss two other measures of weak non-compactness lk(H) and k′(H) for a Fréchet space and provide two quantitative versions of Krein’s theorem for both functions.

Mathematics Subject Classification (2010)

Primary 46A50 Secondary 54C35 


Krein’s theorem Compactness Fréchet space Space of continuous functions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Angosto C., Cascales B.: Measures of weak noncompactness in Banach spaces. Topology Appl. 156, 1412–1421 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    C. Angosto, Distance to spaces of functions, PhD thesis, Universidad de Murcia (2007).Google Scholar
  3. 3.
    C. Angosto and B. Cascales, A new look at compactness via distances to functions spaces, World Sc. Pub. Co. (2008).Google Scholar
  4. 4.
    Angosto C., Cascales B.: The quantitative difference between countable compactness and compactness. J. Math. Anal. Appl. 343, 479–491 (2008)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Angosto C., Cascales B., Namioka I.: Distances to spaces of Baire one functions. Math. Z. 263, 103–124 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    C. Angosto, J. Ka̧kol, and M. López-Pellicer, A quantitative approach to weak compactness in Fréchet spaces and spaces C(X), J. Math. Anal. Appl. 403 (2013), 13–22.Google Scholar
  7. 7.
    Cascales B., Marciszesky W., Raja M.: Distance to spaces of continuous functions. Topology Appl. 153, 2303–2319 (2006)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    M. Fabian et al. Functional Analysis and Infinite-dimensional geometry, CMS Books in Mathematics, Canadian Math. Soc., Springer (2001).Google Scholar
  9. 9.
    M. Fabian et al. A quantitative version of Krein’s theorem, Rev. Mat. Iberoam. 21 (2005), 237–248Google Scholar
  10. 10.
    Granero A. S.: An extension of the Krein-Smulian Theorem. Rev. Mat. Iberoam. 22, 93–110 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Granero A. S., Hájek P., Montesinos V.: Santalucía, Convexity and ω*-compactness in Banach spaces. Math. Ann. 328, 625–631 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Grothendieck A.: Criteres de compacité dans les spaces fonctionnelles généraux. Amer. J. Math. 74, 168–186 (1952)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Khurana S. S.: Weakly compactly generated Fréchet spaces. Int. J. Math. Math. Sci. 2, 721–724 (1979)CrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Carlos Angosto
    • 1
  • Jerzy Ka̧kol
    • 2
  • Albert Kubzdela
    • 3
  • Manuel López-Pellicer
    • 4
  1. 1.Depto. de Matemática Aplicada y EstadisticaUniversidad Politécnica de CartagenaCartagenaSpain
  2. 2.Faculty of Mathematics and InformaticsA. Mickiewicz UniversityPoznanPoland
  3. 3.Institute of Civil EngineeringPoznań University of TechnologyPoznanPoland
  4. 4.Depto. de Matemática Aplicada and IUMPAUniversitat Politècnica de ValènciaValenciaSpain

Personalised recommendations