{
  "id": "2009.07663",
  "version": "v1",
  "published": "2020-09-16T13:04:19.000Z",
  "updated": "2020-09-16T13:04:19.000Z",
  "title": "Integral representation and supports of functionals on Lipschitz spaces",
  "authors": [
    "Ramón J. Aliaga",
    "Eva Pernecká"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We analyze the relationship between Borel measures and continuous linear functionals on the space $\\mathrm{Lip}_0(M)$ of Lipschitz functions on a complete metric space $M$. In particular, we describe continuous functionals arising from measures and vice versa. In the case of weak$^\\ast$ continuous functionals, i.e. members of the Lipschitz-free space $\\mathcal{F}(M)$, measures on $M$ are considered. For the general case, we show that the appropriate setting is rather the uniform (or Samuel) compactification of $M$ and that it is consistent with the treatment of $\\mathcal{F}(M)$. This setting also allows us to give a definition of support for all elements of $\\mathrm{Lip}_0(M)^\\ast$ with similar properties to those in $\\mathcal{F}(M)$, and we show that it coincides with the support of the representing measure when such a measure exists. For a wide class of elements of $\\mathrm{Lip}_0(M)^\\ast$, containing all of $\\mathcal{F}(M)$, we deduce that its members which can be expressed as the difference of two positive functionals admit a Jordan-like decomposition into a positive and a negative part.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-16T13:04:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "46E27",
      "46B40",
      "46B10"
    ],
    "keywords": [
      "lipschitz spaces",
      "integral representation",
      "complete metric space",
      "continuous functionals",
      "similar properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}