 
Summary: Journal of Functional Analysis 242 (2007) 136
www.elsevier.com/locate/jfa
Approximation by smooth functions with no critical
points on separable Banach spaces
D. Azagra 1
, M. JiménezSevilla ,2
Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense,
28040 Madrid, Spain
Received 1 March 2006; accepted 29 August 2006
Available online 17 October 2006
Communicated by G. Pisier
Abstract
We characterize the class of separable Banach spaces X such that for every continuous function
f :X R and for every continuous function :X (0,+) there exists a C1 smooth function g :X R
for which f (x)g(x) (x) and g (x) = 0 for all x 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 Cp, for p = 1,2,...,+. In particular,
we obtain the optimal order of smoothness of the approximating functions with no critical points on the
classical spaces p(N) and Lp(Rn). Some important consequences of the above results are (1) the exis
tence of a nonlinear HahnBanach theorem and the smooth approximation of closed sets, on the classes
