# [Banach] Abstract of a paper by Vladimir Pestov

Dale Alspach alspach at www.math.okstate.edu
Tue Jul 27 10:54:00 CDT 2004

This is an announcement for the paper "Oscillation stability of the
Urysohn metric space" by Vladimir Pestov.

Abstract: We outline general concepts of oscillation stability and
distortion for spaces with action of a topological transformation group,
and survey a number of examples. We observe that the universal Urysohn
metric space $\U$ (viewed as a homogeneous factor-space of its group of
isometries) is oscillation stable, that is, for every bounded uniformly
continuous function $f\colon\U\to\R$ and each $\e>0$ there is an isometric
copy $\U^\prime\subset\U$ of $\U$, such that $f\vert_{\U^\prime}$ is
constant to within $\e$. This stands in marked contrast to the unit sphere
$\s^\infty$ of the Hilbert space $\ell^2$, which is a universal analogue
of $\U$ in the class of spherical metric spaces, but has the distortion
property according to a well-known result by Odell and Schlumprecht.

Archive classification: Functional Analysis

Mathematics Subject Classification: 05C55; 22F30; 43A85; 46B20; 54E35;
54H15

Remarks: 10 pages, LaTeX 2e

The source file(s), osc.tex: 48054 bytes, is(are) stored in gzipped
form as 0407444.gz with size 16kb. The corresponding postcript file has
gzipped size 63kb.

Submitted from: vpest283 at uottawa.ca

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/math.FA/0407444

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http://arXiv.org/abs/math.FA/0407444

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uget 0407444

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