Uniform embeddings of metric spaces and of banach spaces into hilbert spaces
- Cite this article as:
- Aharoni, I., Maurey, B. & Mityagin, B.S. Israel J. Math. (1985) 52: 251. doi:10.1007/BF02786521
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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.