Israel Journal of Mathematics

, Volume 52, Issue 3, pp 251–265

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

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

DOI: 10.1007/BF02786521

Cite this article as:
Aharoni, I., Maurey, B. & Mityagin, B.S. Israel J. Math. (1985) 52: 251. doi:10.1007/BF02786521


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.

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

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