{
  "id": "1005.1050",
  "version": "v5",
  "published": "2010-05-06T17:39:39.000Z",
  "updated": "2010-12-31T16:39:01.000Z",
  "title": "Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces",
  "authors": [
    "D. Azagra",
    "R. Fry",
    "L. Keener"
  ],
  "comment": "Updated version with a sharper result in the Hilbertian case. One thin tube is enough. Some misprints corrected",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\\geq 1$ (depending only on $X$) such that for every Lipschitz function $f:X\\rightarrow\\mathbb{R}$, and every $\\epsilon>0$, there exists a Lipschitz, real analytic function $g:X\\rightarrow\\mathbb{R}$ such that $|f(x)-g(x)|\\leq \\epsilon$ and $\\textrm{Lip}(g)\\leq C\\textrm{Lip}(f)$. This result is new even in the case when $X$ is a Hilbert space. Furthermore, in the Hilbertian case we also show that $C$ can be assumed to be any number greater than 1.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2010-12-31T16:39:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20"
    ],
    "keywords": [
      "real analytic approximation",
      "hilbert space",
      "lipschitz function",
      "real analytic function",
      "separable banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.1050A"
    }
  }
}