Some Remarks on Schauder Bases in Lipschitz Free Spaces

TL;DR

The paper investigates the properties of Schauder bases in Lipschitz free spaces, showing limitations on their existence and behavior in relation to geometric structures like graphs and nets.

Contribution

It demonstrates that retractional Schauder bases in Lipschitz free spaces can have unbounded basis constants and do not exist for certain discrete subsets, revealing fundamental constraints.

Findings

Basis constant increases with radius in graph circle free spaces

No retractional Schauder basis exists for certain discrete subsets of ^2

No retractional unconditional basis exists for free spaces over nets in ^n

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 $\mathcal F(N)$.