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Summary: EXACT FILLING OF FIGURES WITH THE DERIVATIVES OF
SMOOTH MAPPINGS BETWEEN BANACH SPACES
D. AZAGRA(1)
, M. FABIAN(2)
AND M. JIM´ENEZ-SEVILLA(1)
Abstract. We 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 dens X = dens Ln
(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 bound-
ary 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
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