RT Journal Article
T1 On the range of the derivatives of a smooth function betweenBanach spacesy
A1 Azagra Rueda, Daniel
A1 Jiménez Sevilla, María del Mar
A1 Deville, Robert
AB 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-p smooth (Frechet smooth or a continuous G (a) over cap teaux 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 Gateaux case, when X and Y are separable and X is infinite-dimensional, there exists a continuous G (a) over cap teaux smooth function f from X to Y, with bounded support, so that f'(X) = L (X, Y). In the Frechet case, we get that if a Banach space X has a Frechet smooth bump and dens X = dens L(X, Y), then there is a Frechet smooth function f: X --> Y with bounded support so that f'(X) = L(X, Y). Moreover, we see that if X has a C-p smooth bump with bounded derivatives and dens X = dens L-s(m) (X; Y) then there exists another C-p smooth function f : X --> Y so that f((k)) (X) = L-s(k) (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 (Frechet or G (a) over cap teaux) 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.
PB Cambridge University Press
SN 1469-8064
YR 2003
FD 2003-01
LK https://hdl.handle.net/20.500.14352/49769
UL https://hdl.handle.net/20.500.14352/49769
LA eng
NO DGICYT
NO BFM
DS Docta Complutense
RD 30 abr 2024