www.elsevier.com/locate/jmaa 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