{
  "id": "math/0510603",
  "version": "v1",
  "published": "2005-10-27T14:27:41.000Z",
  "updated": "2005-10-27T14:27:41.000Z",
  "title": "Approximation by smooth functions with no critical points on separable Banach spaces",
  "authors": [
    "D. Azagra",
    "M. Jimenez-Sevilla"
  ],
  "comment": "34 pages",
  "categories": [
    "math.FA",
    "math.DG"
  ],
  "abstract": "We characterize the class of separable Banach spaces $X$ such that for every continuous function $f:X\\to\\mathbb{R}$ and for every continuous function $\\epsilon:X\\to\\mathbb(0,+\\infty)$ there exists a $C^1$ smooth function $g:X\\to\\mathbb{R}$ for which $|f(x)-g(x)|\\leq\\epsilon(x)$ and $g'(x)\\neq 0$ for all $x\\in X$ (that is, $g$ has no critical points), as those 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 $p=1,2,..., +\\infty$. In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces $\\ell_p(\\mathbb{N})$ and $L_p(\\mathbb{R}^n)$. Some important consequences of the above results are (1) the existence of {\\em a non-linear Hahn-Banach theorem} and (2) the smooth approximation of closed sets, on the classes of spaces considered above.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-10-27T14:27:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "46T30",
      "58E05",
      "58C25"
    ],
    "keywords": [
      "separable banach space",
      "critical points",
      "smooth function",
      "state sufficient conditions",
      "continuous function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....10603A"
    }
  }
}