{
  "id": "1607.06639",
  "version": "v1",
  "published": "2016-07-22T11:48:21.000Z",
  "updated": "2016-07-22T11:48:21.000Z",
  "title": "Vector lattices and $f$-algebras: the classical inequalities",
  "authors": [
    "Gerard Buskes",
    "Christopher Schwanke"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean (real or complex) vector lattice, from which the Cauchy-Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean (real or complex) $f$-algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove the H\\\"older inequality for weighted geometric mean closed Archimedean (real or complex) $\\Phi$-algebras, improving results by Boulabiar and Toumi. As a consequence, the Minkowski inequality for weighted geometric mean closed Archimedean (real or complex) $\\Phi$-algebras is obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-22T11:48:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "inequality",
      "vector lattice",
      "weighted geometric mean closed archimedean",
      "classical inequalities",
      "sesquilinear maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}