Free Banach spaces and the approximation properties

TL;DR

This paper characterizes metric spaces with free Banach spaces that have the bounded approximation property, using a Lipschitz local reflexivity principle, and shows some free spaces lack the approximation property.

Contribution

Introduces a Lipschitz analogue of local reflexivity to characterize free space approximation properties and provides examples of free spaces failing the approximation property.

Findings

Characterization of metric spaces with free spaces having the bounded approximation property

Existence of compact metric spaces whose free spaces lack the approximation property

Development of a Lipschitz local reflexivity principle

Abstract

We characterize the metric spaces whose free space has the bounded approximation property through a Lipschitz analogue of the local reflexivity principle. We show that there exist compact metric spaces whose free spaces fail the approximation property.