Some Results on Approximation Properties of Lipschitz Maps

We prove thet a Banach space X has the \(\lambda \) $\lambda$ -bounded compact approximation property (respectively, weak \(\lambda \) $\lambda$ -bounded approximation property) if and only if X has the \(\lambda \) $\lambda$ -Lipschitz bounded compact approximation property (respectively, weak \(\lambda \) $\lambda$ -Lipschitz bounded approximation property). Also, it is shown that the dual space of X has the approximation property if and only if for every separable reflexive Banach space Y, the space of finite rank Lipschitz maps from X to Y is dense in the Lipschitz norm in the space of compact operators from X to Y.