Azagra Rueda, DanielFabián, M.Jiménez Sevilla, María Del Mar2023-06-202023-06-2020050008-439510.4153/CMB-2005-045-9https://hdl.handle.net/20.500.14352/49708We establish sufficient conditions on the shape of a set A included in the space Ln s (X; Y ) of the n-linear symmetric mappings between Banach spaces X and Y , to ensure the existence of a Cn-smooth mapping f : X ¡! Y , with bounded support, and such that f(n)(X) = A, provided that X admits a Cn- smooth bump with bounded n-th derivative and densX = densLn(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 Fr´echet smooth case for mappings from Rn to a separable infinite-dimensional Banach space and the Gˆateaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space.engjournal articlehttp://cms.math.ca/cmb/open access517.98Starlike BodiesRangeTheoremBumpAnálisis funcional y teoría de operadores