Some remarks on Schauder bases in Lipschitz free spaces (Q2190718)

The paper deals with the existence of a Schauder basis in ``Lipschitz-free Banach spaces'' \(\mathcal{F}(M)\) (known also as ``Arens-Eells spaces'' or ``transportation cost spaces''). The author provides an example of a discrete metric space \(M\) such that in \(\mathcal{F}(M)\) there exists an ``extensional'' Schauder basis, but there does not exist any ``retractional'' Schauder basis. Moreover, he shows that on \(\mathcal{F}(\mathbb{Z}^d)\) (\(d\geq 2\)) there does not exist a ``retractional'' unconditional Schauder basis (while it is known that a ``retractional'' Schauder basis does exist). The notions of an ``extensional'' and ``retractional'' Schauder basis are summarized below. It is well known that any Schauder basis in a Banach space is given by a collection of projections \((P_n)\). We say the Schauder basis given by the projections \((P_n)\) in a Lipschitz-free space \(\mathcal{F}(M)\) is \textit{extensional} if there are finite subsets \(M_n\subset M\) such that the image of each \(P_n\) is naturally isometric to \(\mathcal{F}(M_n)\) and the adjoints \(P_n^*:\operatorname{Lip}_0(M_n)\to \operatorname{Lip}_0(M)\) provide extensions of Lipschitz functions from \(M_n\) to the space \(M\). This notion is very natural, because as far as the reviewer knows, all the Schauder bases constructed in Lipschitz-free spaces are extensional. Moreover, the basis is \textit{retractional} if the adjoints \(P_n^*\) satisfy \(P_n^*(f) = f\circ r_n\), where \(r_n:M\to M_n\) are certain Lipschitz retractions. Again, this notion is natural as it is often the case that a Schauder basis constructed in \(\mathcal{F}(M)\), where \(M\) is discrete, is retractional.