Smooth extension of functions on a certain class of non-separable Banach spaces

Abstract

Let us consider a Banach space X with the property that every real-valued Lipschitz function f can be uniformly approximated by a Lipschitz, C 1 -smooth function g with Lip ( g ) ⩽ C Lip ( f ) (with C depending only on the space X). This is the case for a Banach space X bi-Lipschitz homeomorphic to a subset of c 0 ( Γ ) , for some set Γ, such that the coordinate functions of the homeomorphism are C 1 -smooth (Hájek and Johanis, 2010 [10]). Then, we prove that for every closed subspace Y ⊂ X and every C 1 -smooth (Lipschitz) function f : Y → R , there is a C 1 -smooth (Lipschitz, respectively) extension of f to X. We also study C 1 -smooth extensions of real-valued functions defined on closed subsets of X. These results extend those given in Azagra et al. (2010) [4] to the class of non-separable Banach spaces satisfying the above property.