 
Summary: EXACT FILLING OF FIGURES WITH THE DERIVATIVES OF
SMOOTH MAPPINGS BETWEEN BANACH SPACES
D. AZAGRA(1)
, M. FABIAN(2)
AND M. JIM´ENEZSEVILLA(1)
Abstract. We establish sufficient conditions on the shape of a set A included in
the space Ln
s (X, Y ) of the nlinear 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 nth derivative and dens X = dens Ln
(X, Y ). For
instance, when X is infinitedimensional, every bounded connected and open set
U containing the origin is the range of the nth derivative of such a mapping.
The same holds true for the closure of U, provided that every point in the bound
ary of U is the end point of a path within U. In the finitedimensional case,
more restrictive conditions are required. We also study the Fr´echet smooth case
