RT Journal Article
T1 Approximation by smooth functions with no critical points on separable Banach spaces
A1 Azagra Rueda, Daniel
A1 Jiménez Sevilla, María del Mar
AB We characterize the class of separable Banach spaces X such that for every continuous function f : X -> Rand for every continuous function epsilon : X -> (0, +infinity) there exists a C-1 smooth function g: X -> R for which vertical bar f(x) - g(x)vertical bar <= epsilon(x) and g'(x) not equal 0 for all x is an element of X (that is, g has no critical points), as those infinite-dimensional Banach spaces X with separable dual X*. We also state sufficient conditions on a separable Banach space so that the function g can be taken to be of class C-p, for p = 1, 2,..., +infinity. In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces l(p)(N) and L-p(R-n). Some important consequences of the above results are (1) the existence of a non-linear Hahn-Banach theorem and the smooth approximation of closed sets, on the classes of spaces considered above; and (2) versions of all these results for a wide class of infinite-dimensional Banach manifolds.
PB Elsevier
SN 0022-1236
YR 2007
FD 2007-06-01
LK https://hdl.handle.net/20.500.14352/49820
UL https://hdl.handle.net/20.500.14352/49820
LA eng
NO Marie Curie Intra-European Fellowship of the European Community, Human Resources andMobility
NO Fellowship of the Secretaría de Estado de Universidades e Investigación (Ministerio de Educación y
DS Docta Complutense
RD 6 may 2024