On Schauder bases in Lipschitz-free spaces
type: Item
instance of: scholarly article
title: On Schauder bases in Lipschitz-free spaces
zbMATH Open document ID: 1319.46010
author: E. Pernecká
author: Petr Hájek
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-01-26T00:00:00Z
review text: For a pointed metric space \(M\), consider the Banach space \(\mathrm{Lip}_0(M)\) of Lipschitz real-valued functions on \(M\) vanishing at \(0\), endowed with the norm defined by the Lipschitz constant. Its unit ball being compact for the pointwise convergence topology, this space is a dual space and a canonical predual of \(\mathrm{Lip}_0(M)\) is \(\mathcal{F}(M)\), the Lipschitz-free space over \(M\). The main result of this paper is that \(\mathcal{F}(\ell_1)\) and \(\mathcal{F}(\mathbb{R}^n)\) have a Schauder basis. More generally, it is proved that the Lipschitz-free space over a product of countably many closed intervals of \(\mathbb{R}\) with endpoints in \(\mathbb{Z}\cup\{-\infty,+\infty\}\), endowed with the distance inherited by the \(\ell_1\)-norm, has a Schauder basis. The method that the authors use is a refinement of the method from \textit{G. Lancien} and \textit{E. Pernecká} [J. Funct. Anal. 264, No. 10, 2323--2334 (2013; Zbl 1291.46017)] to prove the existence of a monotone finite-dimensional Schauder decomposition for \(\mathcal{F}(\ell_1)\) and \(\mathcal{F}(\ell_1^n)\).
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 6390672
zbMATH Keywords: Lipschitz-free space
zbMATH Keywords: Schauder basis
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.02.060
OpenAlex ID: W2086877857
cites work: Q4918199
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DOI: 10.1016/J.JMAA.2014.02.060
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description: scientific article; zbMATH DE number 6390672