Skip to main content
SpringerLink
Log in

Convexity and Smoothness

  • Chapter
Köthe-Bochner Function Spaces

  • 344 Accesses

Abstract

In the first part of this chapter (Section 2.1 and 2.2), we present some basic results about various types of convexity and smoothness conditions that the norm of a Banach space may satisfy. For convexity, we consider strictly convex, uniformly convex, locally uniformly convex, fully rotund, and nearly uniformly convex spaces. Particularly, we present a proof of the Schlumprecht-Odell theorem that shows that every separable reflexive Banach space admits an equivalent 2R norm. On the smoothness, we consider smooth, Fréchet smooth, uniformly Gâteaux smooth and uniformly smooth spaces. In the second part (Section 2.3), we introduce the spreading model, and we show that a Banach space X has the weak Banach-Saks property if and only if X does not have an ℓ1-spreading model.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

2.5 References

  1. G.A. Aleksandrov and I.P. Dimitrov, On the equivalent weakly midpoint locally uniformly rotund renorming of the space l ∞ (in Russian), in Mathatics and Mathematical Education (Sunny Beach, 1985), 189–191.

    Google Scholar 

  2. K.W. Anderson, Midpoint local uniform convexity and other geometric properties of Banach spaces, dissertation, University of Illinois, 1960.

    Google Scholar 

  3. S. Banach, Theory of Linear Operations, North Holland Mathematical Library, vol 38 (1987).

    Google Scholar 

  4. S. Banach and S. Saks, Sur la convergence forte dans les champs L p, Studia Math. 2 (1930), 51–57.

    MATH  Google Scholar 

  5. B. Beauzamy, Espaces d’Interpolation réels: Topologie et Géométrie, Lecture Notes in Math. 666, Springer-Verlag (1978).

    Google Scholar 

  6. B. Beauzamy, Banach-Saks properties and spreading models, Math. Scand. 44 (1979), 357–384.

    MathSciNet  MATH  Google Scholar 

  7. B. Beauzamy, Introduction to Banach Spaces and Their Geometry, Notas de Mathematica, 68, North Holland, Amsterdam (1982).

    Google Scholar 

  8. B. Beauzamy, Propriété de Banach-Saks, Studia Math. 66 (1980) 227–235.

    MathSciNet  MATH  Google Scholar 

  9. B. Beauzamy and J.T. Lapresté, Modèles etalés des espaces de Banach, Travaux ens cours, Hermann, Paris (1984).

    Google Scholar 

  10. J. Bernai and F. Sulliva, Multi-dimensional volumes, superreflexive and normal structure in Banach spaces, Illinois J. Math. 27 (1983), 501–513.

    MathSciNet  Google Scholar 

  11. J. Bourgain, l ∞ /C 0 has no equivalent strictly convex norm, Proc. Amer. Math. Soc. 78 (1980), 225–226.

    MathSciNet  MATH  Google Scholar 

  12. A. Brunei and L. Sucheston, On J-convexity and ergodic super-properties of Banach spaces, Trans. Amer. Math. Soc. 204 (1975), 79–90.

    MathSciNet  Google Scholar 

  13. J.A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396–414.

    Article  MathSciNet  Google Scholar 

  14. D.F. Cudia, Rotundity, Proc. Sympos. Pure Math. Vol 7 (1963) Amer. Math. Soc. Providence, RI, 73–97.

    Google Scholar 

  15. M.M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), 313–317.

    Article  MathSciNet  Google Scholar 

  16. M.M. Day, Uniform convexity in factor and conjugate spaces, Ann. of Math. 45 (1944), 375–385.

    Article  MathSciNet  MATH  Google Scholar 

  17. M.M. Day, Strict convexity and smoothness of norm spaces, Trans. Amer. Math. Soc. 78 (1955), 516–528.

    Article  MathSciNet  MATH  Google Scholar 

  18. M.M. Day, Normed Linear Spaces, Springer-Verlag, New York (1973).

    MATH  Google Scholar 

  19. R. Deville, G. Godefroy, and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific and Technical (1993).

    Google Scholar 

  20. M.M. Day, R.C. James, and S. Swaminatham, Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23 (1971), 1051–1059.

    Article  MathSciNet  Google Scholar 

  21. J. Diestel, Geometry of Banach Spaces: Selected Topics, Lecture Notes in Math. 485, Springer-Verlag, New York (1975).

    Google Scholar 

  22. J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York (1984).

    Book  Google Scholar 

  23. P. Dowling, Z. Hu, and M.A. Smith, MLUR renorming of Banach spaces, Pacific J. Math. 170 (1995) 473–482.

    MathSciNet  MATH  Google Scholar 

  24. K. Fan and I. Glicksburg, Fully convex normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 947–953.

    Article  MathSciNet  MATH  Google Scholar 

  25. K. Fan and I. Glicksburg, Some geometric properties of the spheres in a normed linear spaces, Duke Math. J. 25 (1958), 553–568.

    Article  MathSciNet  MATH  Google Scholar 

  26. N.R. Farnum, The Banach-Saks Theorem in C(S), Can. J. Math. 26 (1974), 91–97.

    Article  MathSciNet  MATH  Google Scholar 

  27. A.L. Garkavi, The best possible net and best possible cross-section of a set in a normed space, Amer. Math. Soc. Transi. 39 (1964), 111–132.

    Google Scholar 

  28. J.R. Giles, Convex Analysis with Application in Differentiation of Convex Functions, Research Notes in Math. 58 Pitman, Boston-London-Melbourne (1982).

    Google Scholar 

  29. G. Godefroy and V. Zizler, Roughness properties of norm on non-Asplund spaces, Michigan Math. J. 38 (1991), 461–466.

    Article  MathSciNet  MATH  Google Scholar 

  30. Z. Hu, W.B. Moors, and M.A. Smith, On a Banach space space without a weak midpoint locally uniformly rotund norm, Bull. Austral. Math. Soc., 56 (1997), 193–196.

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), 743–749.

    Article  MathSciNet  MATH  Google Scholar 

  32. M.I. Kadec, Relation between some properties of convexity of the unit ball of a Banach space, Functional Anal. Appl. 16 (1982), 204–206.

    Google Scholar 

  33. W.A. Kirk, A fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly, 72 (1965), 1004–1006.

    Article  MathSciNet  MATH  Google Scholar 

  34. V.L. Klee, Mappings into normed linear spaces, Fund. Math. 49 (1960-61), 25–34.

    MathSciNet  MATH  Google Scholar 

  35. H. Knaust, E. Odell, and Th. Schlumprecht, On asymptotic structure, the Szlenk index and UKK properties in Banach space, Positivity 3 (1999), 173–199.

    Article  MathSciNet  MATH  Google Scholar 

  36. D.N. Kutzarova and P.K. Lin, Remarks about Schlumprecht space, Proc. Amer. Math. Soc. 128 (2000), 2059–2068.

    Article  MathSciNet  MATH  Google Scholar 

  37. D.N. Kutzarova and S.L. Troyanski, On equivalent lattice norms which are uniform convex or uniform differentiable in every direction in Banach lattices with a weak unit, Sedica Bulgaricae Math. Publ. 9 (1983), 249–262.

    MathSciNet  MATH  Google Scholar 

  38. Bor-Luh Lin and Xin-Tai Yu, On the k-uniform rotund and the fully convex Banach spaces, J. Math. Analysis and Applications 110 (1985), 407–410.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Lindenstrass, Weakly compact sets-their topological properties and the Banach spaces they generate, Sympos. on infinite dimensional topology, Ann. of Math. Studies 69 (1972), 235–275.

    Google Scholar 

  40. A.R. Lovaglia, Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc. 78 (1955), 225–238.

    Article  MathSciNet  MATH  Google Scholar 

  41. D.P. Milman and V.D. Milman, The geometry of nested families with empty intersection, the structure of the unit sphere in a nonreflexive space, Amer. Math. Soc. Transi. 85 (1969), 233–243.

    Google Scholar 

  42. V.D. Milman, Infinite-dimensional geometry of the unit sphere in a Banach space, Soviet Math. Dokl. 8 (1967), 1440–1444.

    Google Scholar 

  43. E. Odell and Th. Schlumprecht, On the richness of the set of p’s in Krivine’s theorem, Oper. Theory: Adv. Appl. 77 (1995), 177–198.

    MathSciNet  Google Scholar 

  44. E. Odell and Th. Schlumprecht, Asymptotic properties of Banach spaces under renormings, J. Amer. Math. Soc. 11 (1998), 175–188.

    Article  MathSciNet  MATH  Google Scholar 

  45. E. Odell and Th. Schlumprecht, A problem on spreading models, J. Funct. Anal. 153 (1998), 249–261.

    Article  MathSciNet  MATH  Google Scholar 

  46. R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer-Verlag (1989).

    Google Scholar 

  47. B.J. Pettis, A proof that every uniformly convex space is reflexive, Duke Math. J. 5 (1939), 249–253.

    Article  MathSciNet  Google Scholar 

  48. J. Radon, Theorie und Anwendungen der absolut additiven Mengen Funktionen, Sitzungsber. Akad. Wiss. Wien 122 (1913), 1295–1438.

    MATH  Google Scholar 

  49. J. Rainwather, Local uniform convexity of Day’s norm on c 0 (Γ), Proc. Amer. Math. Soc. 22 (1969), 335–339.

    MathSciNet  Google Scholar 

  50. M.A. Smith, Some examples concerning rotundity in Banach spaces, Math. Ann. 233 (1978), 155–161.

    Article  MathSciNet  MATH  Google Scholar 

  51. M.A. Smith, A Banach space that is MLUR but not HR, Math. Ann. 256 (1981), 277–279.

    Article  MathSciNet  MATH  Google Scholar 

  52. V.L. Šmulian, On Some geometrical properties of the unit sphere in the space of the type (B), Mat. Sbornik 6 (1938), 77–89.

    Google Scholar 

  53. V.L. Šmulian, Sur la dérivabilité de la norme dans Vespaces de Banach, C.R. Acad. Sci. URSS (Doklady) N.S. 27 (1940), 643–648.

    MATH  Google Scholar 

  54. S.L. Troyanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1971), 173–180.

    MathSciNet  MATH  Google Scholar 

  55. V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.) 87 (1971), 1–35.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Memphis, Memphis, TN, 38152, USA

    Pei-Kee Lin

Authors
  1. Pei-Kee Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this chapter

Cite this chapter

Lin, PK. (2004). Convexity and Smoothness. In: Köthe-Bochner Function Spaces. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8188-3_2

Download citation

  • DOI: https://doi.org/10.1007/978-0-8176-8188-3_2

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6482-8

  • Online ISBN: 978-0-8176-8188-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation