Schauder bases in Lipschitz free spaces over nets in Banach spaces

TL;DR

This paper constructs explicit Schauder bases for Lipschitz free spaces over nets in Banach spaces, providing dimension-independent bases for finite-dimensional spaces and bases for spaces with specific decompositions.

Contribution

It introduces two new explicit constructions of Schauder bases in Lipschitz free spaces over nets, applicable to finite-dimensional spaces and spaces with FDDs, with uniform basis constants.

Findings

Constructed Schauder bases with uniform constants for finite-dimensional Banach spaces.

Established Schauder bases for Lipschitz free spaces over nets in Banach spaces with Schauder bases containing c0.

Provided bases for Lipschitz free spaces over nets in Banach spaces with c0-like FDDs.

Abstract

In the present note we give two explicit constructions (based on a retractional argument) of a Schauder basis for the Lipschitz free space $\mathcal{F}(N)$, over certain uniformly discrete metric spaces $N$. The first one applies to every net $N$ in a finite dimensional Banach space, leading to the basis constant independent of the dimension. The second one applies to grids in Banach spaces with an FDD. As a corollary, we obtain a retractional Schauder basis for the Lipschitz free space $\mathcal{F}(N)$ over a net $N$ in every Banach space $X$ with a Schauder basis containing a copy of $c_0$, as well as in every Banach space with a $c_0$-like FDD.