Lipschitz-free spaces and the metric approximation property

Speaker: Dr Richard Smith (UCD)

Date: Tuesday, September 30, 2014

Time: 3:00pm

Location: Seminar Room, Ag. 1.01

Abstract:

Given a metric space $M$ with distinguished point $0$, the Lipschitz-free space $\mathcal{F}(M)$ is the natural predual of the space of Lipschitz functions that vanish at $0$ (endowed with the Lipschitz norm). The study of these spaces is an emerging area of research. Despite their elementary definition, the linear structure of the spaces $\mathcal{F}(M)$ is still relatively poorly understood: in many cases it is not known whether $\mathcal{F}(M)$ has the approximation property, a finite-dimensional decomposition or a Schauder basis. In this talk we show that for certain subsets $M$ of $\mathbb{R}^N$ (such as all finite-dimensional compact convex sets), the Lipschitz-free space $\mathcal{F}(M)$ has the metric approximation property, independent of the choice of norm on $\mathbb{R}^N$. This contrasts with the fact, proved by Godefroy and Ozawa, that there exist infinite-dimensional compact convex sets $M$ such that $\mathcal{F}(M)$ does not have the approximation property. This is joint work with Eva Perneck&aacute from the Universit&eacute de Franche-omt&eacute 

Series: Analysis