Approximation by Smooth Functions with No Critical Points on Separable Banach Spaces

We characterize the class of separable Banach spaces X such that for every continuous function f : X → R and for every continuous function ε : X → (0,+∞) there exists a C smooth function g : X → R for which |f(x)− g(x)| ≤ ε(x) and g′(x) 6= 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 C, 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 lp(N) and Lp(R ). 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.