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


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 aL 0(μ) space (=the space of the measurable functions on a probability space with convergence in probability). As a result we get thatl p (respectivelyL p (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.


Hilbert Space Banach Space Unit Ball Probability Space Normed Space 
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.


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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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