Rolle's theorem is either false or trivial in infinite-dimensional Banach spaces

Abstract

We prove the following new characterization of $C^p$ (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space $X$ has a $C^p$ smooth (Lipschitz) bump function if and only if it has another $C^p$ smooth (Lipschitz) bump function $f$ such that $f'(x)\neq 0$ 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 $C^p$ 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 $\ell_2$, 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 $\ell_2$ can be uniformly approximated by $C^1$ smooth Lipschiz functions $ψ$ so that the cones generated by the sets of derivatives $ψ'(\ell_{2})$ have empty interior. This implies that there are $C^1$ smooth Lipschitz bumps in $\ell_{2}$ so that the cones generated by their sets of gradients have empty interior.