Approximation by smooth functions with no critical points on separable Banach spaces

Abstract

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.