On Schauder bases in Lipschitz-free spaces
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 \(\mathcal{F}(\ell_1^n)\).
scientific article; zbMATH DE number 6390672
scientific article