Some Remarks on Schauder Bases in Lipschitz Free Spaces

Abstract

We show that the basis constant of every retractional Schauder basis on the Free space of a graph circle increases with the radius. As a consequence, there exists a uniformly discrete subset $M\subset\mathbb{R}^2$ such that $\mathcal F(M)$ does not have a retractional Schauder basis. Furthermore, we show that for any net $ N\subseteq\mathbb{R}^n$ there is no retractional unconditional basis on the Free space $\mathcal F(N)$.