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

Abstract

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.