Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces
type: Item
instance of: scholarly article
title: Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces
zbMATH Open document ID: 0596.46010
DOI: 10.1007/BF02786521
published in: Israel Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 3960262
zbMATH Keywords: positive definite functions
zbMATH Keywords: uniformly homeomorphic to a subset of a Hilbert space
zbMATH Keywords: uniformly embedded
zbMATH Keywords: geometrical properties of Banach spaces
author: Israel Aharoni
author: Bernard Maurey
author: Boris Mityagin
reviewed by: Nassif Ghoussoub
MaRDI profile type: Publication
cites work: Every separable metric space is Lipschitz equivalent to a subset of \(C_0\)
cites work: Uniform embeddings of Banach spaces
cites work: Theory of Reproducing Kernels
cites work: Q5514858
cites work: Functions Positive Definite in the Space C
cites work: On the nonexistence of uniform homeomorphisms between \(L^ p\)-spaces
cites work: On a problem of Smirnov
cites work: Q5515106
cites work: Classical Banach spaces
cites work: Q4136031
cites work: Q3226024
cites work: Espaces de Banach superstables, distances stables et homeomorphismes uniformes
cites work: Metric Spaces and Positive Definite Functions
full work available at URL: https://doi.org/10.1007/bf02786521
OpenAlex ID: W2089706081
description: scientific article; zbMATH DE number 3960262