Advertisement

Journal of Mathematical Sciences

, Volume 180, Issue 6, pp 731–747 | Cite as

The limiting form of the Radon–Nikodym property is true for all Fréchet spaces

  • I. V. Orlov
  • F. S. Stonyakin
Article

Abstract

In this paper, we propose a new limiting form of the Radon–Nikodym property for the Bochner integral. We prove that the limiting form holds for an arbitrary Fr´echet space as opposed to an ordinary Radon–Nikodym property. We consider some applications in linear and nonlinear analysis.

Keywords

Banach Space Radon Convex Compact Convex Space Inductive Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. D. Arvanitakis and A. Aviles, “Some examples of continuous images of Radon–Nikodym compact spaces,” arXiv:0903.0653v1 [math.GN], 1–11 (2009).Google Scholar
  2. 2.
    Yu. M. Berezansky, Z. Gr. Sheftel, and G. F. Us, Functional Analysis, 1, Birkhäuser Verlag, Basel–Boston–Berlin (1995).Google Scholar
  3. 3.
    Q. Bu, G. Buskes, and L. Wei-Kai, “The Radon–Nikodym property for tensor products of banach lattices II,” Positivity, 12, 45-54 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    N. D. Chakraborty and Sk. Jaker Ali, “Type II-Λ-weak Radon–Nikodym property in a Banach space associated with a compact metrizable abelian group,”Extracta Math, 23, No. 3, 201–216 (2008).MathSciNetzbMATHGoogle Scholar
  5. 5.
    J. Cheeger and B. Kleiner, “Characterization of the Radon–Nikodym property in terms of inverse limits,” arXiv:0706.3389v3 [math.FA], 1–12 (2008).Google Scholar
  6. 6.
    J. Cheeger and B. Kleiner, “Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon–Nikodym property,” arXiv:0808.3249v1 [math.MG], 1–17 (2008).Google Scholar
  7. 7.
    G. Chi, “A geometric characterization of Fréchet spaces with the RNT,” Proc. Amer. Math. Soc., 48, 371–380 (1975).MathSciNetzbMATHGoogle Scholar
  8. 8.
    W. J. Davis, “The Radon–Nikodym property,” Seminaire d’analyse fonctionelle (Polytechnique), exp no. 0, 1–12 (1973-1974).Google Scholar
  9. 9.
    J. Diestel and J. J. Uhl, Vector Measures, Amer. Math. Soc., Providence (1977).zbMATHGoogle Scholar
  10. 10.
    N. Dunford and B. J. Pettis, “Linear operations on summable functions,” Trans. Amer. Math. Soc., 47, 323–392 (1940).MathSciNetCrossRefGoogle Scholar
  11. 11.
    N. Dunford and J. T. Schwartz, Linear Operators. General Theory [in Russian], Inostrannaya Literatura, Moscow (1962).Google Scholar
  12. 12.
    D. Gilliam, “Geometry and the Radon–Nikodym theorems in strict Mackey convergence spaces,” Pacific J. Math., 65, No. 2, 353–364 (1976).MathSciNetzbMATHGoogle Scholar
  13. 13.
    E. Hille and R. S. Phillips, Functional Analysis and Semigroups[in Russian], Inostrannaya Literatura, Moscow (1962).Google Scholar
  14. 14.
    S. Moedomo and J. J. Uhl, “Radon–Nikodym theorems for the Bochner and Pettis integrals,” Pacific J. Math., 38, No. 2, 531–536 (1971).MathSciNetzbMATHGoogle Scholar
  15. 15.
    I. V. Orlov, “Convergence almost everywhere as convergence in the nonlinear inductive scale of locally convex spaces,” Uch. Zap. Tavrich. Univ., Ser. Math., 14 (53), 75–80 (2001).Google Scholar
  16. 16.
    I. V. Orlov, “Formula of finite increments for maps into inductive scales of spaces,” Mat. Fiz. Anal. Geom., 8, No. 4, 419–439 (2001).MathSciNetzbMATHGoogle Scholar
  17. 17.
    I. V. Orlov and F. S. Stonyakin, “Compact subdifferentials: the formula of finite increments and related topics” Sovrem. Mat. Fundam. Napravl., 34, 121–138 (2009).Google Scholar
  18. 18.
    I. V. Orlov and F. S. Stonyakin, “Compact variation, compact subdifferential and indefinite Bochner integral,” Methods Funct. Anal. Topology, 15, No. 1, 74–90 (2009).MathSciNetzbMATHGoogle Scholar
  19. 19.
    I. V. Orlov and F. S. Stonyakin, “Strong compact properties of the mappings and K-property of Radon–Nikodym,” Methods Funct. Anal. Topology, To appear.Google Scholar
  20. 20.
    R. S. Phillips, “On weakly compact subsets of a Banach space,” Amer. J. Math., 65, No. 3, 108–136 (1943).MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    M. A. Rieffel, “The Radon–Nikodym theorem for the Bochner integral,” Trans. Amer. Math. Soc., 131, 466–487 (1968).MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    H. H. Shaefer, Topological Vector Spaces, Macmillan, New York–London (1966).Google Scholar
  23. 23.
    V. A. Trenogin, B. M. Pisarevskii, and T. S. Soboleva, Tasks and Exercises on Functional Analysis [in Russian], Nauka, Moscow (1984).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Taurida National V. Vernadsky UniversitySimferopolUkraine

Personalised recommendations