Smooth approximation of Lipschitz functions on Riemannian manifolds

We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous ε :M →(0,+∞), and for every positive number r > 0, there exists a C∞ smooth Lipschitz function g :M →R such that |f (p) − g(p)| ε(p) for every p ∈ M and Lip(g) Lip(f )+r. Consequently, every separable Riemannian manifold is uniformly bumpable.We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville–Godefroy–Zizler’s smooth variational principle. © 2006 Elsevier Inc. All rights reserved.