Advertisement

Stability Properties I

  • Pei-Kee Lin

Abstract

Let X be a Banach space. Recall that a unit vector x in X is said to be an extreme point if for any y, z ε B(X), x = 1/2(y+z) implies x = y = z. A unit vector x in X is said to be a smooth point of X if there is a unique x* ε S(X*) such that (x*, x) = 1.

Keywords

Banach Space Function Space Extreme Point Unit Ball Stability Property 
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.

4.5 References

  1. [1]
    Ch. Castaing and R. Pluciennik, Denting points in Köthe-Bochner spaces, Set-valued Analysis, 2 (1994), 439–458.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    J. Cerdà, H. Hudzik, and M. Mastyło, Geometric properties in Köthe-Bochner spaces, Math. Proc. Camb. Phil. Soc. 120 (1996), 521–533.MATHCrossRefGoogle Scholar
  3. [3]
    S. Chen and B.-L. Lin, Strongly extreme points in Köthe-Bochner spaces, Rocky Mountain J. Math. 27 (1997), 1055–1063.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    W. Deeb and R. Khalil, Smooth points of vector valued function spaces, Rocky Mountain J. 24 (1994), 505–512.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    P. Greim, An extremal vector-valued L p-function taking no extremal vector as values, Proc. Amer. Math. Soc. 84 (1982), 65–68.MathSciNetMATHGoogle Scholar
  6. [6]
    P. Greim, Strongly exposed points in Bochner L p-spaces, Proc. Amer. Math. Soc. 88 (1983), 81–84.MathSciNetMATHGoogle Scholar
  7. [7]
    P. Greim, A note on strong extreme and strongly exposed points in Bochner L p-spaces, Proc. Amer. Math. Soc. 93 (1985), 65–66.MathSciNetMATHGoogle Scholar
  8. [8]
    W. Hensgen, Exposed points in Lebesgue-Bochner and Hardy-Bochner spaces, J. Math. Analysis and Applications 198 (1996), 780–795.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Z. Hu and B.-L. Lin, RNP and CPCP in Lebesgue-Bochner function spaces, Illinois J. Math. 37 (1993), 329–347.MathSciNetMATHGoogle Scholar
  10. [10]
    Z. Hu and B.-L. Lin, A characterization of weak* denting points in Bochner L p-spaces, Rocky Mountain J. Math. 24 (1994), 997–1008.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Z. Hu and B.-L. Lin, Strongly exposed points in Lebesgue-Bochner function spaces, Proc. Amer. Math. Soc. 120 (1994), 1159–1165.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Z. Hu and B.-L. Lin, Extremal structure of the unit ball of L p (μ, X)*, J. Math. Analysis and Applications 200 (1996), 567–590.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    H. Hudzik and M. Mastyło, Strongly extreme points in Köthe-Bochner spaces, Rocky Mountain J. Math. 23 (1993), 899–909.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    J.A. Johnson, Extreme measurable selections, Proc. Amer. Math. Soc. 44 (1974), 107–112.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    J.A. Johnson, Strongly exposed points in L p (μ, X), Rocky Mountain J. Math. 10 (1980), 517–519.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    A. Kamińska and B. Turett, Rotundity in Köthe spaces of vector-valued functions, Can. J. Math. 41 (1989), 659–675.MATHCrossRefGoogle Scholar
  17. [17]
    I.E. Leonard and K. Sundaresan, Smoothness and duality in L p (E, μ), J. Math. Ana. Appl. 46 (1974), 513–522.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    I.E. Leonard and K. Sundaresan, Geometry of Lebesgue-Bochner function spaces: smoothness, Trans. Amer. Math. Soc. 198 (1974), 229–251.MathSciNetMATHGoogle Scholar
  19. [19]
    B.-L. Lin and P.K. Lin, Property (H) in Lebesgue-Bochner function spaces, Proc. Amer. Math. Soc. 95 (1985), 581–584.MathSciNetMATHGoogle Scholar
  20. [20]
    B.-L. Lin and P.K. Lin, Denting points in Bochner L p-spaces, Proc. Amer. Math. Soc. 97 (1986), 629–633.MathSciNetMATHGoogle Scholar
  21. [21]
    P.K. Lin, Stability of some properties in Köthe-Bochner function spaces, in Function Spaces, the fifth conference, Pure and Applied Math, vol 213, Marrel Dekker, Inc. (2000), 347–357.Google Scholar
  22. [22]
    P.K. Lin and H. Sun, Extremal structure in Köthe function spaces, J. Math. Analysis and Applications 218 (1998), 136–154.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    R. Pluciennik, On characterization of strongly extreme points in Köthe-Bochner spaces, Rocky Mountain J. Math. 27 (1997), 307–315.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    W. Schachermayer, For a Banach space isomorphic to its square the Radon-Nikodým property and the Krein-Milman property are equivalent, Studia Math. 85 (1985), 329–339.MathSciNetGoogle Scholar
  25. [25]
    M.A. Smith, Strongly extreme points in L p (μ, X), Rocky Mountain J. Math. 16 (1986), 1–5.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    M.A. Smith, Rotundity and extremity in l p (X) and L p (μ, X), in Geometry of Normed Linear Spaces, Contemporary Math. 52 (1986), 143–162.CrossRefGoogle Scholar
  27. [27]
    M.A. Smith and B. Turett, Rotundity in Lebesgue-Bochner function spaces, Trans. Amer. Math. Soc. 257 (1980), 105–118.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    K. Sundaresan, Extreme points of the unit cell in Lebesgue-Bochner function spaces, Colloq. Math. 22 (1970), 111–119.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