Abstract. We introduce and study the Kunen-Shelah properties KS i , i = 0, 1, . . . , 7. Let us highlight some of our results for a Banach space X: (1)

Abstract. We show that if Y is a separable subspace of a Banach space X such that both X and the quotient X/Y have C p -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 C p -smooth Lipschitz function F :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 p -smooth (Lipschitz) perturbed extension of f ? That is, for a givenSuch 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 p -smooth partitions of unity, and also when Y has a C p -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 1 -smooth bump function, uniformly continuous functions can be approximated by C 1 -smooth Lipschitz functions, uniformly on bounded sets.

Abstract. We prove that if X is an infinite-dimensional Banach space with C p smooth partitions of unity then X and X \ K are C p diffeomorphic for every weakly compact set K ⊂ X.