论文信息 - Uniform embeddings of metric spaces and of banach spaces into hilbert spaces

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

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.

B. Mityagin | B. Maurey | I. Aharoni | Israel Aharoni

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