Extensions of Lipschitz functions and Grothendieck’s bounded approximation property

Abstract

A metric compact space \(M\) is seen as the closure of the union of a sequence \((M_n)\) of finite \(\epsilon_n\)-dense subsets. Extending to \(M\) (up to a vanishing uniform distance) Banach-space valued Lipschitz functions defined on \(M_n\), or defining linear continuous near-extension operators for real-valued Lipschitz functions on \(M_n\), uniformly on \(n\) is shown to be equivalent to the bounded approximation property for the Lipschitz-free space \(\mathcal{F}(M)\) over \(M\). Several consequences are spelled out.