Siberian Mathematical Journal

, Volume 50, Issue 1, pp 162–166 | Cite as

The Kreps-Yan theorem for Banach ideal spaces

  • D. B. Rokhlin


Consider a closed convex cone C in a Banach ideal space X on some measure space with σ-finite measure. We prove that the fulfilment of the conditions CX + = {0} and C⊃−X + guarantees the existence of a strictly positive continuous functional on X whose restriction to C is nonpositive.


Kreps-Yan theorem Banach ideal space σ-finite measure cone separation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kantorovich L. V. and Akilov G. P., Functional Analysis, Pergamon Press, Oxford; New York (1982).zbMATHGoogle Scholar
  2. 2.
    Aliprantis C. D. and Border K. C., Infinite-Dimensional Analysis. A Hitchhikerĩs Guide, Springer-Verlag, Berlin (2006).Google Scholar
  3. 3.
    Väth M., Ideal Spaces, Springer-Verlag, Berlin (1997) (Lecture Notes in Math.; 1664).zbMATHGoogle Scholar
  4. 4.
    Jouini E., Napp C., and Schachermayer W., “Arbitrage and state price deflators in a general intertemporal framework,”; J. Math. Econom., 41, No. 6, 722–734 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kreps D. M., “Arbitrage and equilibrium in economies with infinitely many commodities,”; J. Math. Econom., 8, No. 1, 15–35 (1981).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Yan J. A., “Caractérisation dĩune classe dĩensembles convexes de L 1 ou H 1,”; in: Séminare de Probabilites XIV, Springer-Verlag, Berlin, 1980, pp. 220–222 (Lecture Notes in Math.; 784).Google Scholar
  7. 7.
    Rokhlin D. B., “The Kreps-Yan theorem for L ,”; Internat. J. Math. Math. Sci., 2005, No. 17, 2749–2756 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cascales D., Namioka I., Orihuela J., and Raja M., “Banach spaces and topology (I),”; in: Encyclopedia of General Topology, Elsevier, New York, 2003, pp. 449–453.CrossRefGoogle Scholar
  9. 9.
    Cassese G., “Yan theorem in L with applications to asset pricing,”; Acta Math. Appl. Sinica (English Ser.), 23, No. 4, 551–562 (2007).CrossRefMathSciNetGoogle Scholar
  10. 10.
    Corson H. H., “The weak topology of a Banach space,”; Trans. Amer. Math. Soc., 101, No. 1, 1–15 (1961).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bogachev V. I., Fundamentals of Measure Theory. Vol. 1 [in Russian], Nauchno-Issled. Tsentr “Regulyarnaya i Khaoticheskaya Dinamika”;, Moscow; Izhevsk (2003).Google Scholar
  12. 12.
    Brannath W. and Schachermayer W., “A bipolar theorem for L +0(Ω, P),”; in: Séminaire de Probabilites XXXIII, Springer-Verlag, Berlin, 1999, pp. 349–354 (Lecture Notes in Math.; 1709).CrossRefGoogle Scholar
  13. 13.
    Rokhlin D. and Schachermayer W., “A note on lower bounds of martingale measure densities,”; Illinois J. Math., 50, No. 4, 815–824 (2006).zbMATHMathSciNetGoogle Scholar
  14. 14.
    Lozanovskiĭ G. Ya., “On some Banach lattices,”; Sibirsk. Mat. Zh., 10, No. 3, 584–599 (1969).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Southern Federal UniversityRostov-on-DonRussia

Personalised recommendations