Advertisement

Mathematische Annalen

, Volume 335, Issue 3, pp 687–715 | Cite as

Biorthogonal systems and quotient spaces via baire category methods

  • Stevo Todorcevic
Article

Abstract

We show that every Banach space X of density smaller that the Baire category number Open image in new window admits a quotient with a long Schauder basis that can be taken of length ω1 if X is not separable. So, assuming that the Baire category number Open image in new window does not take its minimal possible value, a Banach space X is separable if and only if all biorthogonal systems of X are countable.

Keywords

Banach Space Category Method Quotient Space Category Number Baire Category 
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.
    Banach, S.: Théorie des opérations linéaires, Warszawa 1932Google Scholar
  2. 2.
    Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis. Amer. Math. Soc., Providence 2000Google Scholar
  3. 3.
    Bell, M., Ginsburg, J., Todorcevic, S.: Countable spread of exp Y and λY. Topology and its Appl. 14, 1–12 (1982)zbMATHCrossRefGoogle Scholar
  4. 4.
    Borwein, J.M., Vanderwerff, J.D.: Banach spaces that admit support sets. Proc. Amer. Math. Soc. 124, 751–755 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Davis, W.J., Johnson, W.B.: On the existence of fundamental and total bounded biorthogonal systems in Banach spaces. Studia Math. 45, 173–179 (1972)Google Scholar
  6. 6.
    Engelking, R.: General Topology, Heldermann Verlag Berlin 1989Google Scholar
  7. 7.
    Fabian, M., Habala, P., Hajek, P., Montesinos, V., Pelant, J., Zizler, V.: Functional Analysis and Infinite-Dimensional Geometry. CMS Books in Mathematics, Springer-Verlag, 2001Google Scholar
  8. 8.
    Finet, C., Godefroy, G.: Biorthogonal systems and big quotient spaces. Contemp. Math. 85, 275–283 (1989)Google Scholar
  9. 9.
    Foreman, M., Magidor, M., Shelah, S.: Martin's maximum, saturated ideals and non-regular ultrafilters. Ann. Math. 127, 1–47 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fremlin, D.H.:Consequences of Martin's axioms. Cambridge Univ. Press 1984Google Scholar
  11. 11.
    Frontisi, J.: Lissité et dialité dans les espaces de Banach. Thése de Dectorat, Univ. Paris VI, 1996Google Scholar
  12. 12.
    Godefroy, G., Louveau, A.: Axioms of determinacy and biorthogonal systems. Israel J. Math. 67, 109–116 (1989)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Grodefroy, G., Talagrand, M.: Espaces de Banach representables. Israel J. Math. 41, 321–330 (1982)MathSciNetGoogle Scholar
  14. 14.
    Godun, B.V., Kadets, M.: Banach spaces without complete minimal systems. Funk. Analiz i Ego Pril. 14, 67–68 (1980)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Johnson, W.B.: No infinite dimensional P space admits a Markushevich basis. Proc. Amer. Math. Soc. 28, 467–468 (1970)CrossRefGoogle Scholar
  16. 16.
    Johnson, W.B.: E-mail message of November 16, 2005Google Scholar
  17. 17.
    Johnson, W.B., Rosenthal, H.P.: On w*-basic sequences and their applications in the study of Banach spaces. Studia Math. 43, 77–92 (1972)MathSciNetGoogle Scholar
  18. 18.
    Koszmider, P.: A problem of Rolewics about Banach spaces that admit support sets, preprint 2004Google Scholar
  19. 19.
    Kunen, K.: Products of S-spaces, note of May 1975Google Scholar
  20. 20.
    Kunen, K.: On hereditarily Lindelof Banach spaces, note of July 1980Google Scholar
  21. 21.
    Kutzarova, D.: Convex sets containing only support points in Banach spaces with an uncountable minimal system. C.R. l'Acad. Bulg. Sci., 39, 13–14 (1986)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Lazar, A.J.: Points of support for closed convex sets. Illinois J. Math. 25, 302–305 (1981)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces, I, Springer-Verlag, Berlin, 1977Google Scholar
  24. 24.
    Mujica, J.: Separable quotients of Banach spaces. Revista Mat. Univ. Complut. 10, 299–330 (1997)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Montesinos, V.: Solution to a problem of S. Rolewicz. Studia Math. 81, 65–69 (1985)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Negrepontis, S.: Banach spaces and topology, Handbook of Set-Theoretic topology. In: Kunen, K., Vaughan, J.E., (eds), North-Holland, 1984, pp. 1045–1142Google Scholar
  27. 27.
    Pelczynski, A.: Some problems on basis in Banach and Fréchet space. Israel J. Math. 2, 132–138 (1964)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Plichko, A.N.: Fundamental biorthogonal systems and projective bases in Banach spaces. Math. Zametki 34, 473–476 (1983)Google Scholar
  29. 29.
    Plichko, A.N.: Banach spaces without fundamantal biorthogonal systems. Soviet Math. Dokl. 254(4), 978–801 (1980)Google Scholar
  30. 30.
    Rolewicz, S.: On convex sets containing only points of support. Comment. Math., Prace Math. Special issue 1, 279–281 (1978)MathSciNetGoogle Scholar
  31. 31.
    Singer, I.: Bases in Banach spaces II, Springer-Verlag, Berlin, 1981Google Scholar
  32. 32.
    Shelah, S.: Uncountable constructions for BA, e.g. groups and Banach spaces. Israel J. Math. 51, 273–297 (1985)zbMATHGoogle Scholar
  33. 33.
    Todorcevic, S.: Forcing positive partition relations. Trans. Amer. Math. Soc. 280, 703–720 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Todorcevic, S.: Directed sets and cofinal types. Trans. Amer. Math. Soc. 290, 711–723 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Todorcevic, S.: Partition problems in topology, Amer. Math. Soc. Providence 1989Google Scholar
  36. 36.
    Todorcevic, S.: Irredundant sets in Boolean algebras. Trans. Amer. Math. Soc. 339, 35–44 (1990)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Todorcevic, S.: Some applications of S and L combinatorics. Ann. New York Acad. Sci.,705, 130–167 (1993)Google Scholar
  38. 38.
    Todorcevic, S.: Basis problems in combinatorial set theory, Proc. International Congress of Mathematicians Berlin 1998, Docum. Math. Extra Volume ICM 1998. II. 43–52Google Scholar
  39. 39.
    Todorcevic, S.: A generic function space Open image in new window(K) with no support set, in preparationGoogle Scholar
  40. 40.
    Wojtaszczyk, P.: A theorem on convex sets related to the abstract Pontriagin maximality principle. Bull. Acad. Polon Sci 21, 93–95 (1973)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Université Paris 7 -C.N.R.S., UMR 7056Paris Cedex 05France

Personalised recommendations