{
  "id": "math/0002108",
  "version": "v1",
  "published": "2000-02-14T17:48:23.000Z",
  "updated": "2000-02-14T17:48:23.000Z",
  "title": "Rolle's theorem is either false or trivial in infinite-dimensional Banach spaces",
  "authors": [
    "Daniel Azagra",
    "Mar Jimenez-Sevilla"
  ],
  "comment": "20 pages",
  "categories": [
    "math.FA",
    "math.DG"
  ],
  "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 $\\psi$ so that the cones generated by the sets of derivatives $\\psi'(\\ell_{2})$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-02-14T17:48:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "58B99"
    ],
    "keywords": [
      "infinite-dimensional banach space",
      "bump function",
      "empty interior",
      "hilbert space",
      "smooth lipschiz functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......2108A"
    }
  }
}