Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces (Q1078475)

It is proved, using positive definite functions, that a normed space X is uniformly homeomorphic to a subset of a Hilbert space, if and only if X is (linearly) isomorphic to a subspace of a \(L_ 0(\mu)\) space \((=\) the space of the measurable functions on a probability space with convergence in probability). As a result we get that \(\ell_ p\) (respectively \(L_ p(0,1))\), \(2<p<\infty\), 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.