Robert Deville Laboratoire d’Analyse et Géométrie Université Bordeaux I 351 Cours de la Libération 33400 Talence Cedex FRANCE

On the range of the derivative of a Gateaux-smooth mapping between Banach spaces

Abstract : We consider the following problem : Given two separable real Banach spaces X and Y, does there exist a Lipschitz continuous and bounded mapping from X to Y such that f is Gateaux differentiable at each point of X, and, for each x, y ∈ X we have ||f′(x) − f′(y)|| > 1 whenever x is different from y ? We show that this is not always possible (for instance this is not possible if dim(Y) = 1 ), but there are Banach spaces X and Y where this construction can be performed, for instance when X = Y is the separable Hilbert space, or when X = ℓ^{1} and dim(Y) = 2. This answers a question of Gilles Godefroy raised during the seventh edition of the International Conference on Infinite Dimensional Analysis held in Madrid in 2001.