{
  "id": "1602.07158",
  "version": "v1",
  "published": "2016-02-23T14:11:16.000Z",
  "updated": "2016-02-23T14:11:16.000Z",
  "title": "On the infimum of certain functionals",
  "authors": [
    "Biagio Ricceri"
  ],
  "categories": [
    "math.FA",
    "math.OC"
  ],
  "abstract": "In this note, in particular, we establish the following result: Let $X$ be a real Banach space, $\\varphi\\in X^*\\setminus \\{0\\}$ and $\\psi:X\\to {\\bf R}$ a Lipschitzian functional with Lipschitz constant equal to $\\varphi\\|_X^{*}$. Then, we have $$\\max\\left\\{\\inf_{x\\in X}(\\varphi(x)+\\psi(x)),\\inf_{x\\in X}(\\varphi(x)-\\psi(x))\\right\\}=\\inf_{x\\in X}(\\varphi(x)+|\\psi(x)|)$$ and $$\\liminf_{\\|x\\|\\to +\\infty}(\\varphi(x)+|\\psi(x)|)=\\inf_{x\\in X}(\\varphi(x)+|\\psi(x)|)\\ .$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-23T14:11:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real banach space",
      "lipschitz constant equal",
      "lipschitzian functional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}