On the coincidence of the canonical embeddings of a metric space into a Banach space
type: Item
instance of: scholarly article
title: On the coincidence of the canonical embeddings of a metric space into a Banach space
zbMATH Open document ID: 1188.46014
published in: Journal of Mathematical Sciences (New York)
publication date: 2009-11-13T00:00:00Z
review text: Let \((X,\rho)\) be a metric space and \(F(X)\) be the space of finitely supported signed measures on \(X\) with zero total charge (that is, taking zero values on constant functions). The author considers two norms on \(F(X)\). One of them, the Kantorovich-Rubinstein norm, is defined as \(\|\mu\|_{KR}=\sup |\int f\,d\mu|\), where the supremum is over all \(1\)-Lipschitz functions \(f:X\to\mathbb{R}\). The second, the author calls it the Hausdorff-Kuratowski norm, is defined as \(\|\mu\|_{HK}=\sup_{x\in X}|\int \rho(x,y)\,d\mu(y)|\). The main result of the paper says that the norms \(\|\mu\|_{KR}\) and \(\|\mu\|_{HK}\) are different for finite metric spaces \(X\) with at least \(5\) points. The main motivation for the paper is that the measures are related to two canonical ways of embedding a metric space into a Banach space. Reviewer's remarks: (1) The completion of the normed space \((F(X),\|\mu\|_{KR})\) is also known as the Arens-Eells space [see \textit{R.\,F.\thinspace Arens} and \textit{J.\,Eells}, Pac.\ J.\ Math.\ 6, 397--403 (1956; Zbl 0073.39601) and \textit{N.\,Weaver}, Lipschitz algebras.\ Singapore: World Scientific (1999; Zbl 0936.46002)]. Closely related objects are the Lipschitz-free space [see \textit{G.\,Godefroy} and \textit{N.\,J .\,Kalton}, Stud.\ Math.\ 159, No.\,1, 121--141 (2003; Zbl 1059.46058)] and the free Banach space [see \textit{V.\,G.\thinspace Pestov}, Funct.\ Anal.\ Appl.\ 20, 70--72 (1986); translation from Funkts.\ Anal.\ Prilozh.\ 20, No.\,1, 81--82 (1986; Zbl 0635.22005)]. (2) The embedding \(x\mapsto\rho(x,\cdot)\) into the space \(C(X)\), which the author calls the Hausdorff-Kuratowski embedding, is also called the Fréchet embedding, at least for finite metric spaces [see \textit{J.\,Matoušek}, ``Lectures on discrete geometry'' (Graduate Texts in Mathematics 212; New York/NY:\ Springer) (2002; Zbl 0999.52006)].
Mathematics Subject Classification ID: 46B85
zbMATH DE Number: 5632971
zbMATH Keywords: Banach space
zbMATH Keywords: isometric embedding
zbMATH Keywords: metric space
author: Pavel B. Zatitskiy
reviewed by: Mikhail Ostrovskii
MaRDI profile type: Publication
full work available at URL: s10958-009-9422-2
OpenAlex ID: W1970780471
cites work: Linearly rigid metric spaces and the embedding problem
DOI: 10.1007/S10958-009-9422-2
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description: scientific article; zbMATH DE number 5632971