Approximation properties in terms of Lipschitz maps
We investigate some approximation properties of Banach spaces which are described in terms of Lipschitz maps. First, we present characterizations of the Lipschitz approximation property, and prove that a Banach space $X$ has the approximation property whenever the Lipschitz-free space over $X$ has this property. Furthermore, we obtain a Lipschitz version of Grothendieck’s characterization of the classical approximation property. Second, we introduce the Lipschitz weak $\lambda $-bounded approximation property and show that it implies the classical weak $\lambda $-bounded approximation property. Finally, several equivalent formulations of the Lipschitz weak $\lambda $-bounded approximation property are obtained.