Exact Filling of Figures with the Derivatives of Smooth Mappings between Banach Spaces

We establish sufficient conditions on the shape of a set A included in the space Lns (X, Y ) of the n-linear symmetric mappings between Banach spaces X and Y , to ensure the existence of a C-smooth mapping f : X −→ Y , with bounded support, and such that f (X) = A, provided that X admits a Csmooth bump with bounded n-th derivative and dens X = densL(X, Y ). For instance, when X is infinite-dimensional, every bounded connected and open set U containing the origin is the range of the n-th derivative of such a mapping. The same holds true for the closure of U , provided that every point in the boundary of U is the end point of a path within U . In the finite-dimensional case, more restrictive conditions are required. We also study the Fréchet smooth case for mappings from R to a separable infinite-dimensional Banach space and the Gâteaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space.