A note on the range of the derivatives of analytic approximations of uniformly continuous functions on co

Abstract

This paper is a contribution to the body of results concerning the size of the set of derivatives of differentiable functions on a Banach space. The results so far have consisted of examples of highly differentiable bump functions (or functions approximating a given continuous function) whose set of derivatives either is surprisingly small or has a given shape. The paper under review treats the case of analytic smoothness. The main result states that every uniformly continuous function on c0 (more generally on a space with property (K)) can be approximated by a real-analytic function whose set of derivatives is contained in T p>0 lp. This is a significant step forward, as analytic functions are substantially harder to deal with than C1 smooth ones. Indeed, a local perturbation of an analytic function necessarily changes the values of the function everywhere. Also of particular value is the quite precise and elegant description of the set of derivatives of the approximating function.