Stability Properties II

  • Pei-Kee Lin


One of natural questions about Köthe-Bochner function spaces is the following: Problem. Let E be a Köthe function space and X a Banach space. When does the space E(X) contain a copy (or a complemented copy) of c 0, l 1, or l ?


Banach Space Stability Property Banach Lattice Polish Space Finite Sequence 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

5.8 References

  1. [1]
    C. Abbott, E. Bator, R. Bilou, and P. Lewis, Weak precompactness, strong boundedness and weak complete continuity, Math. Camb. Phil. Soc. 108 (1990), 325–335.MATHCrossRefGoogle Scholar
  2. [2]
    D.J. Aldous, Unconditional bases and martingales in L P (F), Math. Proc. Camb. Phil. Soc. 85 (1979), 117–123.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    J. Batt and W. Hiermeyer, On compactness in L p (μ, X) in the weak topology and in the topology σ(L p (μ, X), L q (μ, X*), Math. Z. 182 (1983), 409–423.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    B. Beauzamy and J.T. Lapresté, Modèles etalés des espaces de Banach, Travaux ens cours, Hermann, Paris (1984).Google Scholar
  5. [5]
    F. Bombai, On l 1 subspaces of Orlicz vector-valued function spaces, Math. Proc. Camb. Phil. Soc. 101 (1987), 107–112.CrossRefGoogle Scholar
  6. [6]
    J. Bourgain, On the Dunford-Pettis property, Proc. Amer. Math. Soc. 81 (1981), 265–272.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    J. Bourgain, An averaging result for c 0-sequences, Bull. Soc. Math. Belg. Ser. B. 30 (1978), 83–87.MathSciNetMATHGoogle Scholar
  8. [8]
    J. Bourgain, An averaging result for l 1-sequences and applications to weakly conditionally compact sets in L 1 X, Israel J. Math. 32 (1979), 289–298.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    D.L. Burkholder, Martingale and singular integrals in Banach spaces, in Handbook of the Geometry of Banach Spaces, vol. 1, edited by W. B. Johnson and J. Lindenstrass, Elsevier, (2001) pp. 233–269.Google Scholar
  10. [10]
    P. Cembranos, The weak Banach-Saks property on L p (μ, E), Math. Proc. Camb. Phil. Soc. 115 (1994), 283–290.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    P. Cembranos and J. Mendoza, Banach Spaces of Vector-Valued Functions, Lecture Notes in Math. 1676 Springer-Verlag, Berlin-Heidelberg (1997).Google Scholar
  12. [12]
    S. Díaz, Weak compactness in L 1 (μ, X), Proc. Amer. Math. Soc. 124 (1996), 2685–2693.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    S. Díaz, Complemented copies of c 0 in L (μ, X), Proc. Amer. Math. Soc. 120 (1994) 1167–1172.MathSciNetMATHGoogle Scholar
  14. [14]
    S. Díaz, Complemented copies of l 1 in L (μ, X), Rocky Mountain J. Math. 27 (1997), 779–784.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    S. Díaz and A. Fernandez, Reflexivity in Banach lattices, Archiv der Math. 63 (1994), 549–552.MATHCrossRefGoogle Scholar
  16. [16]
    J. Diestel, The Dunford-Pettis property, Contemp. Math. 2 (1980), 15–60.MathSciNetCrossRefGoogle Scholar
  17. [17]
    J. Diestel, Sequences and Senes in Banach Spaces, Graduate Texts in Math. vol. 92, Springer-Verlag (1984).Google Scholar
  18. [18]
    J. Diestel, W.M. Ruess, and W. Schachermayer, Weak compactness in L l (μ, X), Proc. Amer. Math. Soc. 118 (1993), 447–453.MathSciNetMATHGoogle Scholar
  19. [19]
    J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge University Press 1995.Google Scholar
  20. [20]
    P.N. Dowling, A stability property of a class of Banach spaces not containing c 0, Canadian Math. Bull. 35 (1992), 56–60.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    G. Emmanuele, On complemented copy of c 0 in L p (μ, X), 1 ≤ p < ∞, Proc. Amer. Math. Soc. 104 (1988), 785–786.MathSciNetMATHGoogle Scholar
  22. [22]
    S. Guerre, La propriété de Banach-Saks ne passe pas de E à L 2 (E) d’après J. Bourgain, Séminaire d’Analyse Fonctionnelle, Ecole Polytechnique Paris, (1979/1980).Google Scholar
  23. [23]
    R. Haydon, A nonreflexive Grothendieck space that does not contain l , Israel J. Math. 40 (1981) 65–73.MathSciNetCrossRefGoogle Scholar
  24. [24]
    R. Haydon, An unconditional result about Grothendieck spaces, Proc. Amer. Math. Soc. 100 (1987), 511–516.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    R. Haydon, M. Levy, and E. Odell, On sequences without weak* convergent convex block subsequences, Proc. Amer. Math. Soc. 100 (1987), 94–98.MathSciNetMATHGoogle Scholar
  26. [26]
    J. Hoffmann-Jorgensen, Sums of independent Banach space valued random variable, Studia Math. 52 (1974), 159–185.MathSciNetGoogle Scholar
  27. [27]
    N.J. Kalton, E. Saab and P. Saab, L p(X), 1 ≤ p < ∞, has the property (u) whenever X does, Bull. Sci. Math. Paris. 115 (1991), 369–377.MathSciNetMATHGoogle Scholar
  28. [28]
    S. Kwapien, On Banach spaces containing C 0 Studia Math. 52 (1974), 187–188.MathSciNetMATHGoogle Scholar
  29. [29]
    D. Leung and F. Räbiger, Complemented copies of c 0 l -sums of Banach spaces, Illnois J. Math. 34 (1990), 52–58.MATHGoogle Scholar
  30. [30]
    J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Mathematics 338, Springer-Verlag (1973).Google Scholar
  31. [31]
    B. Maurey, Système de Haar, Séminaire Maurey-Schwartz, 1974-1975, École Polytechnique, Paris (1975).Google Scholar
  32. [32]
    J. Mendoza, Complemented copies of l 1 in L p(μ, E), Math. Proc. Camb. Phil. Soc. 111 (1992), 531–534.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    J. Mendoza, Copies of classical sequence spaces in vector-valued function Banach spaces, in Function Spaces-the second conference, Lecture Notes in Pure and Applied Math. 172, (ed. K. Jarosz) Marcel Dekker (1995) 311–320.Google Scholar
  34. [34]
    M. Nowak, Conditional weak compactness in vector valued function spaces, Proc. Amer. Math. Soc. 129 (2001), 2947–2953.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    A. Pełczynski, On Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. 10 (1962), 641–648.MATHGoogle Scholar
  36. [36]
    G. Pisier, Un exemple concernant la super-réflexivité, Séminaire Maurey-Schwartz (1974-1975), 326–350.Google Scholar
  37. [37]
    G. Pisier, Une propriété de stabilité de la classe des espaces ne contenant pas l 1, C. R. Acad. Sci. Paris Sér. A 286 (1978), 747–749.MathSciNetMATHGoogle Scholar
  38. [38]
    N. Randrianantoanina, Complemented copies of l 1 and PełczyńskVs property (V*) in Bochner function spaces, Canad. J. Math. 48 (1996), 625–640.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    Y. Raynaud, Sous-espaces l r et géométrie des espaces L p (L q) et L ø, C. R. Acad. Sc. Paris, 301 (1985) 299–302.MathSciNetMATHGoogle Scholar
  40. [40]
    Y. Raynaud, Sur les sous-espaces de L p (L q), Séminaire d’Analyse Fonctionelle 1984/1985, Universités Paris-VI et VII (1986), 49–71.Google Scholar
  41. [41]
    E. Saab and P. Saab, On stability problems of some properties in Banach spaces, in Function Spaces, Lecture Notes in Pure and Applied Mathematics 136 (ed. K. Jarosz), Marcel Dekker (1992) 367–394.Google Scholar
  42. [42]
    W. Schachermayer, The Banach-Saks property is not L 2-hereditary, Israel J. Math. 40 (1981), 340–344.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    M.A. Smith and B. Turett, Rotundity in Lebesgue-Bochner function spaces, Trans. Amer. Math. Soc. 257 (1980), 105–118.MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    M. Talagrand, Un nouveau C(K) qui possède la propriété Grothendieck, Israel J. Math. 37 (1980), 181–191.MathSciNetMATHCrossRefGoogle Scholar
  45. [45]
    M. Talagrand, La propnété de Dunford-Pettis dans C(K,E) et L 1(E), Israel, J. Math. 44 (1983), 317–321.MathSciNetMATHCrossRefGoogle Scholar
  46. [46]
    M. Talagrand, Weak Cauchy sequences in L 1(E), Amer. J. Math. 106 (1984), 703–724.MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    M. Talagrand, Quand Vespace des measures à variation bornée est-il faiblemente séquentiellement complet? Proc. Amer. Math. Soc. 90 (1984), 285–288.MathSciNetMATHGoogle Scholar
  48. [48]
    A. Ülger, Weak compactness in L 1(μ, X), Proc. Amer. Math. Soc. 113 (1991), 143–149.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Pei-Kee Lin
    • 1
  1. 1.Department of MathematicsUniversity of MemphisMemphisUSA

Personalised recommendations