On smooth extensions of vector-valued functions defined on closed subsets of Banach spaces

Abstract

Let and be Banach spaces, a closed subset of and a mapping . We give necessary and sufficient conditions to obtain a smooth mapping such that , when either (i) and are Hilbert spaces and is separable, or (ii) is separable and is an absolute Lipschitz retract, or (iii) and with , or (iv) and with , where is any separable Banach space with a -finite measure space.