Rolle’s Theorem Is Either False or Trivial in Infinite-dimensional Banach Spaces

We prove the following new characterization of C (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a C smooth (Lipschitz) bump function if and only if it has another C smooth (Lipschitz) bump function f such that f ′(x) 6= 0 for every point x in the interior of the support of f (that is, f does not satisfy Rolle’s theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. As a by-product of the proof of this result we also obtain other useful characterizations of C smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer’s fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces. Finally, we study the structure of the set of gradients of bump functions in the Hilbert space l2, and as a consequence of the failure of Rolle’s theorem in infinite dimensions we get the following result. The usual norm of the Hilbert space l2 can be uniformly approximated by C smooth Lipschiz functions ψ so that the cones generated by the sets of derivatives ψ′(l2) have empty interior. This implies that there are C 1 smooth Lipschitz bumps in l2 so that the cones generated by their sets of gradients have empty interior.