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Isometric Imbedding of Ultrametric Spaces in the Lebesgue Spaces
by
Alex J. Lemin
Department of Mathematics, Moscow State University of Civil Engineering, Moscow, Russia
The goal of the lecture is to relate the theory of ultrametric spaces to Lebesgue theory. Let m be the Lebesgue measure on the real line R , L(R) be a space of all measurable subsets (of finite measure) of R up to zero-sets. For any measurable A and B, we set d(A, B)=m(ADB), where ADB is a symmetrical difference. We call L(R) the Lebesgue space. It is complete, separable, convex, homogeneous, and not locally compact metric space.
THEOREM 1, [1]. Every separable ultrametric space can be isometrically imbedded in the Lebesgue space L(R).
COROLLARY (I. Gelfand). Every separable ultrametric space can be isometrically imbedded in the space L1(R) of Lebesgue integrable functions on R.
In early 1970-s, Prof. Sergei Nikolski [2] stated the question on an imbedding of any separable ultrametric space in the spaces Lp(R) of Lebesgue integrable functions on L(R) with the norm ||f(x)||p=(ò|f(x)|pdx)1/p. A. Timan [2] gave a partial answer for a certain class of countable ultrametric spaces for p > 1.
THEOREM 2, [1]. Every separable ultrametric space can be isometrically embedded in the space Lp(R) for any p > 0.
We give an axiomatic definition of the Lebesgue space L(R), describe its metric properties, and compare these with geometric properties of Hilbert space. We discuss the problem of imbedding of ultrametric spaces in arbitrary Banach spaces.
References
1. A.J.Lemin. Isometric embedding of ultrametric (non-Archimedean)
spaces in Hilbert space and Lebesgue space, in "p-adic Functional
Analysis" (Lecture Notes in Math), Marcel Dekker, 2001
2. A.F.Timan. On isometric embedding of certain countable ultrametric
spaces in the Lp-spaces, - in "Proceedings of the Mathematical
Institute of the USSR Ac. Sc.", 134 (1975). Collection of articles
dedicated to Sergey Mihailovich Nikolski on the occasion of his 70th
birthday. Moscow, 1975, 314-326 (in Russian).
Date received: May 13, 2001
Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagw-29.