Perturbed Smooth Lipschitz Extensions of Uniformly Continuous Functions on Banach Spaces

We show that if Y is a separable subspace of a Banach space X such that both X and the quotient X/Y have Cp-smooth Lipschitz bump functions, and U is a bounded open subset of X, then, for every uniformly continuous function f : Y ∩U → R and every ε > 0, there exists a Cp-smooth Lipschitz function F : X → R such that |F (y)− f(y)| ≤ ε for every y ∈ Y ∩U . If we are given a separable subspace Y of a Banach space X and a continuous (resp. Lipschitz) function f : Y → R, under what conditions can we ensure the existence of a C-smooth (Lipschitz) perturbed extension of f? That is, for a given ε > 0, does there exist a C-smooth (Lipschitz) function F : X → R so that |F (y) − f(y)| ≤ ε for all y ∈ Y ? Such an F will be called a smooth perturbed extension of f . Of course there are several conditions under which the answer is “yes” in a trivial way (for the non-Lipschitz case). For instance, when X has C-smooth partitions of unity, and also when Y has a C-smooth bump function and the subspace Y is complemented in X , such perturbed extensions F are easily proved to exist. However, in the Lipschitz case, or when the space X does not have smooth partitions of unity and Y is not complemented in X , it is not quite clear what the answer is. These questions are interesting in the theory of Banach spaces because, for instance, while trying to prove a theorem, one might be able to construct a continuous (resp. Lipschitz) function f with some nice properties, but defined only on a certain separable subspace Y of X , and then one might want to obtain a smooth (resp. and Lipschitz) function F defined on the whole of X that behaves on Y almost the same way as f does. In this paper we will try to give a solution to the above question in the Lipschitz case. This problem is clearly related to the question concerning uniform approximation of Lipschitz functions by smooth Lipschitz functions on infinite-dimensional spaces, which has remained unasked and open until recent times: in [7] R. Fry has shown that, on any separable Banach space with a C-smooth bump function, uniformly continuous functions can be approximated by C-smooth Lipschitz functions, uniformly on bounded sets. Received by the editors January 26, 2003. 2000 Mathematics Subject Classification. Primary 46B20. c ©2004 American Mathematical Society Reverts to public domain 28 years from publication