On the Range of the Derivatives of a Smooth Function between Banach Spaces

We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spaces X and Y to ensure the existence of a C smooth (Fréchet smooth or a continuous Gâteaux smooth) function f from X onto Y such that f vanishes outside a bounded set and all the derivatives of f are surjections. In particular we deduce the following results. For the Gâteaux case, when X and Y are separable and X is infinite-dimensional, there exists a continuous Gâteaux smooth function f from X to Y , with bounded support, so that f ′(X) = L(X, Y ). In the Fréchet case, we get that if a Banach space X has a Fréchet smooth bump and densX = densL(X, Y ), then there is a Fréchet smooth function f : X −→ Y with bounded support so that f ′(X) = L(X, Y ). Moreover, we see that if X has a C smooth bump with bounded derivatives and densX = densLs (X;Y ) then there exists another C smooth function f : X −→ Y so that f (X) = Ls (X;Y ) for all k = 0, 1, ..., m. As an application, we show that every bounded starlike body on a separable Banach space X with a (Fréchet or Gâteaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space X∗. In the non-separable case, we prove that X has such property if X has smooth partitions of unity.