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

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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 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.

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Authors and Affiliations

  1. Department of Mathematics, Jerusalem College of Technology, 21 Havaad Haleumi St., Jerusalem, Israel

    I. Aharoni

  2. U.E.R. de Mathematique et Informatique, Universite Paris VII, 2 Place Jussieu, 75251, Paris Cedex 05, France

    B. Maurey

  3. Department of Mathematics, Ohio State University, 43210, Columbus, OH, USA

    B. S. Mityagin

Authors
  1. I. Aharoni
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  2. B. Maurey
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  3. B. S. Mityagin
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Additional information

Partially supported by the National Science Foundation, Grant MCS-79-03322.

Partially supported by the National Science Foundation, Grant MCS-80-06073.

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Aharoni, I., Maurey, B. & Mityagin, B.S. Uniform embeddings of metric spaces and of banach spaces into hilbert spaces. Israel J. Math. 52, 251–265 (1985). https://doi.org/10.1007/BF02786521

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  • DOI: https://doi.org/10.1007/BF02786521

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