Schauder basis in Lipschitz free spaces over nets of $\mathcal{L}_\infty$-spaces

TL;DR

This paper constructs a Schauder basis for Lipschitz free spaces over nets in separable infinite dimensional -spaces, providing new examples of Banach spaces with this property that do not contain c_0.

Contribution

It introduces a new construction of Schauder bases in Lipschitz free spaces over nets in -spaces, including the first example in a space not containing c_0.

Findings

Constructed Schauder basis for  spaces over nets.

First example of such a basis in a space without c_0.

Uses retractional argument for the construction.

Abstract

In the present note we give a construction (based on a retractional argument) of a Schauder basis for the Lipschitz free space $\mathcal{F}(N)$, over a net $N$ in any separable infinite dimensional $\mathcal{L}_\infty$-space $X$. In particular, this yields the first example of an infinite dimensional Banach space $X$ not containing $c_0$ with such a property.