ISRAEL JOURNAL OF MATHEMATICS, Vol. 52, No. 3, 1985 UNIFORM EMBEDDINGS OF METRIC SPACES AND OF BANACH SPACES INTO HILBERT SPACES BY I. AHARONP ¢ B. MAUREY ~ AND B. S. MITYAGIN c'" ~Department of Mathematics, Jerusalem College of Technology, 21 Havaad Haleumi St., Jerusalem, Israel; bUniversite Paris VII, U.E.R. de Mathematique et Informatique, 2 Place Jussieu, 75251 Paris Cedex 05, France ; ~Department of Mathematics, Ohio State University, Columbus, 0H43210, USA ABSTRACT It is proved using positive definite functions that a normed space X is unifomly homeomorphic to a subset of a Hilbert space, if and only if X is (linearly) isomorphic to a subspace of a L,,(/z) space (= the space of the measurable functions on a probability space with convergence in probability). As a result we get that lp (respectively L,(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. 1. Introduction The origin of this note is the investigation of uniform embeddings of Banach spaces in Hilbert spaces. We say that two metric spaces X, Y are uniformly homeomorphic if there is a 1-1 map T from X onto Y, such that T and T ' are uniformly continuous. We say that X is uniformly embedded in Y, if X is uniformly homeomorphic to a subset of Y. In [7] Enflo proved that Co, and in fact every normed space which contains l"~ (linearly) uniformly, is not uniformly embedded in a Hilbert space. (This answers negatively a problem raised by Smirnov (quoted in [8]) whether every separable Banach space is uniformly embedded in a Hilbert space.) Looking for other spaces, it was proved in [2] that t Partially supported by the National Science Foundatio n, Grant MCS-79-03322. " Partially supported by the National Science Foundation, Grant MCS-80-06073. Received November 1, 1984 251