Israel Journal of Mathematics

, Volume 52, Issue 3, pp 251–265 | Cite as

Uniform embeddings of metric spaces and of banach spaces into hilbert spaces

  • I. Aharoni
  • B. Maurey
  • B. S. Mityagin
Article

Abstract

It is proved using positive definite functions that a normed spaceX is unifomly homeomorphic to a subset of a Hilbert space, if and only ifX is (linearly) isomorphic to a subspace of aL0(μ) space (=the space of the measurable functions on a probability space with convergence in probability). As a result we get thatlp (respectivelyLp(0, 1)), 2<p<∞, is not uniformly embedded in a bounded subset of itself. This answers negatively the question whether every infinite dimensional Banach space is uniformly homeomorphic to a bounded subset of itself. Positive definite functions are also used to characterize geometrical properties of Banach spaces.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. Aharoni,Every separable Banach space is Lipschitz equivalent to a subset of c 0, Isr. J. Math.19 (1974), 284–291.CrossRefMathSciNetGoogle Scholar
  2. 2.
    I. Aharoni,Uniform embeddings of Banach spaces, Isr. J. Math.27 (1977), 174–179.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    N. Aronszajn,Theory of reproducing kernels, Trans. Am. Math. Soc.68 (1950), 337–404.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. Bretagnolle, D. Dacunha-Castelle and J. L. Krivine,Lois stables et espaces L, Ann. Inst. Henri Poincare, Sect. B2 (1966), 231–259.MathSciNetGoogle Scholar
  5. 5.
    S. J. Einhorn,Functions positive definite in the space C, Am. Math. Monthly75 (1968), 393.CrossRefMathSciNetGoogle Scholar
  6. 6.
    P. Enflo,On the nonexistence of uniform homeomorphisms between L p-spaces, Ark. Mat.8 (1969), 103–105.CrossRefMathSciNetGoogle Scholar
  7. 7.
    P. Enflo,On a problem of Smirnov, Ark. Mat.8 (1969), 107–109.CrossRefMathSciNetGoogle Scholar
  8. 8.
    E. A. Gorin,On uniformly topological embedding of metric spaces in Euclidean and in Hilbert spaces, Uspehi Mat. Nauk14 (1959), 5 (89), 129–134 (Russian).MATHMathSciNetGoogle Scholar
  9. 9.
    E. Hewitt and K. A. Ross,Abstract Harmonic Analysis 1, Springer-Verlag, Berlin, 1963.Google Scholar
  10. 10.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces, Lectures Notes in Mathematics338, Springer-Verlag, Berlin, 1973.MATHGoogle Scholar
  11. 11.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces 1,Sequence Spaces, Springer-Verlag, Berlin, 1977.Google Scholar
  12. 12.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces 2,Function Spaces, Springer-Verlag, Berlin, 1977.Google Scholar
  13. 13.
    M. Loeve,Probability Theory, Van Nostrand, New York, 1955.MATHGoogle Scholar
  14. 14.
    S. Mazur,Une remarque sur l’homeomorphie des champs fonctionnels, Studia Math.1 (1929), 83–85.Google Scholar
  15. 15.
    E. H. Moore,On properly positive Hermitian matrices, Bull. Am. Math. Soc.23 (1916), 59, 66-67.Google Scholar
  16. 16.
    Y. Raynaud,Espaces de Banach superstables, distances stables et homeomorphismes uniformes, Isr. J. Math.44 (1983), 33–52.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    I. J. Schoenberg,Metric spaces and positive definite functions, Trans. Am. Math. Soc.44 (1938), 522–536.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1985

Authors and Affiliations

  • I. Aharoni
    • 1
  • B. Maurey
    • 2
  • B. S. Mityagin
    • 3
  1. 1.Department of MathematicsJerusalem College of TechnologyJerusalemIsrael
  2. 2.U.E.R. de Mathematique et InformatiqueUniversite Paris VIIParis Cedex 05France
  3. 3.Department of MathematicsOhio State UniversityColumbusUSA

Personalised recommendations