Skip to main content

Convexity and differentiability of norms

  • Chapter
  • First Online:
Geometry of Banach Spaces-Selected Topics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 485))

  • 1050 Accesses

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight 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.

References

  1. S. Banach, Theorie des operations lineaires, Monografje Matematyczne, Vol. 1, Warzawa, 1932.

    Google Scholar 

  2. L. P. Belluce and W. A. Kirk, Fixed-point theorems for families of contraction mappings, Pac. J. Math., 18 (1966), 213–217.

    Article  MathSciNet  MATH  Google Scholar 

  3. -, Non-expansive mappings and fixed points in Banach spaces, Ill. J. Math., 11 (1967), 474–479.

    MathSciNet  MATH  Google Scholar 

  4. L. P. Belluce, W. A. Kirk, and E. F. Steiner, Normal structure in Banach spaces, Pac. J. Math., 26 (1968), 433–440.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J., 1 (1935), 169–172.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. S. Brodskii and D. P. Milman, On the center of a convex set, Dokl. Akad. Nauk (SSSR), 59 (1948), 837–840.

    MathSciNet  Google Scholar 

  7. F. E. Browder, Fixed point theorems for non-compact mappings in Hilbert space, Proc. Nat. Acad. Sci. (USA), 53 (1965), 1272–1276.

    Article  MathSciNet  MATH  Google Scholar 

  8. -, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. (USA), 54 (1965), 1041–1044.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. O. Carlsson, Orthogonality in normed linear spaces, Ark. f. Mat., 4 (1961), 297–318.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. A. Clarkson, Uniformly convex spaces, Trans. AMS, 40 (1936), 396–414.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. F. Cudia, Rotundity, Proc. Sympos. Pure Math. 7 (1963), AMS Providence, RI, pp. 73–97.

    MATH  Google Scholar 

  12. -, The geometry of Banach spaces, Smoothness. Trans. AMS, 110 (1964), 284–314.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. AMS, 47 (1941), 313–317.

    Article  MathSciNet  MATH  Google Scholar 

  14. -, Some more uniformly convex spaces, Bull. AMS, 47 (1941), 504–507.

    Article  MathSciNet  MATH  Google Scholar 

  15. -, Uniform convexity, III. Bull. AMS, 49 (1943), 745–750.

    Article  MathSciNet  MATH  Google Scholar 

  16. -, Uniform convexity in factor and conjugate spaces, Ann. of Math. (2), 45 (1944), 374–385.

    Article  MathSciNet  MATH  Google Scholar 

  17. -, Strict convexity and smoothness of normed spaces, Trans. AMS, 78 (1955), 516–528.

    Article  MathSciNet  MATH  Google Scholar 

  18. -, Every L-space is isomorphic to a strictly convex space, Proc. AMS, 8 (1957), 415–417.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. R. DeMarr, Common fixed points for commuting contraction mappings, Pac. J. Math., 13 (1963), 1139–1141.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Diestel and B. Faires, On vector measures, Trans. Amer. Math. Soc., 198 (1974), 253–271.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Dixmier, Sur un théorème de Banach, Duke Math. Jour., 15 (1948), 1057–1071.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Edelstein, A theorem on fixed points under isometries, Amer. Math. Monthly, 70 (1963), 298–300.

    Article  MathSciNet  MATH  Google Scholar 

  24. -, On nonexpansive mappings of Banach spaces, Proc. Cambridge Phil. Soc., 60 (1964), 439–447.

    Article  MathSciNet  MATH  Google Scholar 

  25. K. Fan and I. Glicksberg, Some geometrical properties of the spheres in a normed linear space, Duke Math. Jour., 25 (1958), 553–568.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. R. Fortet, Remarques sur les espaces uniformément convexes, Bull. Soc. Math. France, 69 (1941), 23–46.

    MathSciNet  MATH  Google Scholar 

  27. A. L. Garkavi, The best possible net and the best possible cross-section of a set in a normed linear space, Izv. Akad. Nauk. SSSR, Ser. Mat., 26 (1962), 87–106; Amer. Math. Soc. Translations, Ser. 2, 39 (1964), 111–132.

    MathSciNet  Google Scholar 

  28. J. R. Giles, Classes of semi-inner-product spaces, Trans. AMS, 129 (1967), 436–446.

    Article  MathSciNet  MATH  Google Scholar 

  29. -, On a characterization of differentiability of the norm of a normed linear space, Jour. Aust. Math. Soc., 12 (1971), 106–114.

    Article  MathSciNet  MATH  Google Scholar 

  30. V. I. Gurarii and N. I. Gurarii, Bases in uniformly convex and uniformly flattened Banach spaces, Math. USSR Izvestija, 5 (1971), 220–225.

    Article  MathSciNet  MATH  Google Scholar 

  31. J. R. Holub, On the metric geometry of ideals of operators on Hilbert space, Math. Ann. 201 (1973), 157–163.

    Article  MathSciNet  MATH  Google Scholar 

  32. R. C. James, Orthogonality in normed linear spaces, Duke Math. Jour., 12 (1945), 291–302.

    Article  MathSciNet  MATH  Google Scholar 

  33. R. C. James, Bases and reflexivity of Banach spaces, Ann. Math., 52 (1950), 518–527.

    Article  MathSciNet  MATH  Google Scholar 

  34. S. Karlin, Bases in Banach spaces, Duke Math. J., 15 (1948), 971–985.

    Article  MathSciNet  MATH  Google Scholar 

  35. 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 

  36. V. Klee, Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc., 74 (1953), 10–43.

    Article  MathSciNet  MATH  Google Scholar 

  37. -, Some new results on smoothness and rotundity in normed linear spaces, Math. Annalen, 139 (1959), 51–63.

    Article  MathSciNet  MATH  Google Scholar 

  38. -, Mappings into normed linear spaces, Fund. Math., 49 (1960), 25–34.

    MathSciNet  MATH  Google Scholar 

  39. G. Köthe, Topological Vector Spaces. Berlin: Springer-Verlag, 1969.

    MATH  Google Scholar 

  40. E. Leonard and K. Sundaresan, Smoothness in Lebesgue-Bochner function spaces and the Radon-Nikodým theorem, to appear.

    Google Scholar 

  41. J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 3 (1963), 139–148.

    Article  MathSciNet  MATH  Google Scholar 

  42. -, Weakly compact sets, their topological properties and Banach spaces they generate, Proc. Symp. Infinite Dimensional Topology, 1967, Ann. of Math Studies, 69 (1972), 235–273.

    MathSciNet  Google Scholar 

  43. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. Springer-Verlag Lecture Note Series, 338 (1973), Berlin-Heidelberg-New York.

    Google Scholar 

  44. A. R. Lovaglia, Locally uniformly convex Banach spaces, Trans. AMS, 78 (1955), 225–238.

    Article  MathSciNet  MATH  Google Scholar 

  45. S. Mazur, Uber knovexe Mengen in linearen normierten Raumen, Studia Math., 4 (1933), 70–84.

    MATH  Google Scholar 

  46. D. P. Milman, On some criteria for the regularity of spaces of the type (B), Dokl. Akad. Nauk SSSR, 20 (1938), 243–246.

    MATH  Google Scholar 

  47. Z. Opial, Nonexpansive and monotone mappings in Banach spaces, Lecture notes from January, 1967 lectures given at Center for Dynamical Systems at brown University.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  49. M. M. Rao, Smoothness in Orlicz spaces I, II. Proc. Amsterdam Acad. Sci., 28 (1965), 671–690.

    MATH  Google Scholar 

  50. A. F. Ruston, A note on convexity of Banach spaces, Proc. Cambridge Philos. Soc., 45 (1949), 157–159.

    Article  MathSciNet  MATH  Google Scholar 

  51. I. Singer, personal communication.

    Google Scholar 

  52. V. L. Šmulyan, On some geometrical properties of the unit sphere in the space of the type (B). (Russian). Mat. Sbornik, 6 (1939), 77–94.

    MathSciNet  Google Scholar 

  53. -, Sur la derivabilité de la norme dans l'espace de Banach, Doklady (CR Aad. Sci. URSS), 27 (1940), 643–648.

    MATH  Google Scholar 

  54. V. L. Šmulyan, Sur la structure de la sphere unitaire dans l'espace de Banach, Math. Sbornik, 9 (51) (1941), 545–561.

    MathSciNet  Google Scholar 

  55. K. Sundaresan, Orlicz spaces isomorphic to strictly convex spaces, Proc. AMS, 17 (1966), 1353–1356.

    Article  MathSciNet  MATH  Google Scholar 

  56. K. Yosida, Functional Analysis. Springer-Verlag, Berlin-Heidelberg-New York, 1960.

    MATH  Google Scholar 

  57. V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math., 87 (1971), 5–33.

    MathSciNet  MATH  Google Scholar 

Download references

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1975 Springer-Verlag

About this chapter

Cite this chapter

Diestel, J. (1975). Convexity and differentiability of norms. In: Geometry of Banach Spaces-Selected Topics. Lecture Notes in Mathematics, vol 485. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082081

Download citation

  • DOI: https://doi.org/10.1007/BFb0082081

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-07402-1

  • Online ISBN: 978-3-540-37913-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics