Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces

Let X be a separable Banach space with a separating polynomial. We show that there exists C 1 (depending only on X) such that for every Lipschitz function f : X→R, and every ε > 0, there exists a Lipschitz, real analytic function g : X→R such that |f (x)−g(x)| ε and Lip(g) C Lip(f ). This result is new even in the case when X is a Hilbert space. Furthermore, in the Hilbertian case we also show that C can be assumed to be any number greater than 1. © 2011 Elsevier Inc. All rights reserved.