Abstract
We prove, among other things, that a Lipschitz (or uniformly continuous) mapping f: X-> Y can be approximated (even in a fine topology) by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a separable C(K) space (resp. super-reflexive space). Further, we show how smooth approximation of Lipschitz mappings is closely related to a smooth approximation of C^1-smooth mappings together with their first derivatives.
As a corollary we obtain new results on smooth approximation of C^1-smooth mappings together with their first derivatives.