Lipschitz free spaces and the approximation property

Abstract

Approximation properties are fundamental in Banach space theory, orig-inating from Grothendieck's work on finite-rank operators. Lipschitz-free spaces extend linear methods to metric settings by linking Lipschitz map-pings with Banach space theory. A key result of Godefroy and Kalton shows the equivalence between the Lipschitz and classical bounded approximation properties.

This thesis studies approximation properties in Lipschitz-free spaces. We establish new characterizations of the Lipschitz bounded approximation prop-erty, prove transfer results between Lipschitz and linear approximations, and give finite-rank reformulations.