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

We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ℒ(X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f′(X)=A. Whenever f is only assumed Gâteaux differentiable, new phenomena appear: for instance, there exists a mapping f from ℓ1(ℕ) into ℝ2, which is bounded, Lipschitz-continuous, and so that for all x,y∈ℓ1(ℕ), if x≠y, then ‖f′(x)−f′(y)‖>1