Lipschitz Embeddings of Metric Spaces into $c_0$

zbMATH Open1424.46033arXiv1612.02025MaRDI QIDQ4631953

Author name not available (Why is that?)

Publication date: 24 April 2019

Abstract: Let $M$ be a separable metric space. We say that $f=(f_n):Mo{c}_0$ is a good-$lambda$-embedding if, whenever $x,yinM$, $xey$ implies $d(x,y)leVertf(x)-f(y)Vert$ and, for each $n$, $Lip(f_n)<lambda$, where $Lip(f_n)$ denotes the Lipschitz constant of $f_n$. We prove that there exists a good-$lambda$-embedding from $M$ into $c_0$ if and only if $M$ satisfies an internal property called $pi(lambda)$. As a consequence, we obtain that for any separable metric space $M$, there exists a good-$2$-embedding from $M$ into $c_0$. These statements slightly extend former results obtained by N. Kalton and G. Lancien, with simplified proofs.

Full work available at URL: https://arxiv.org/abs/1612.02025

zbMATH Keywords

metric spacesLipschitz embeddingsBanach space $c_0$

Mathematics Subject Classification ID

Special maps on metric spaces (54E40) Isometric theory of Banach spaces (46B04) Embedding (54C25) Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science (46B85)

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