{
  "id": "1512.04050",
  "version": "v1",
  "published": "2015-12-13T13:19:08.000Z",
  "updated": "2015-12-13T13:19:08.000Z",
  "title": "Real analyticity of composition is shy",
  "authors": [
    "Seppo I. Hiltunen"
  ],
  "comment": "5 pages, AmS-LaTeX",
  "categories": [
    "math.FA"
  ],
  "abstract": "Dahmen and Schmeding have obtained the result that although the smooth Lie group $G$ of real analytic diffeomorphisms $\\mathbb S^{\\,1.}\\to\\mathbb S^{\\,1.}$ has a compatible analytic manifold structure, it does not make $G$ a real analytic Lie group since the group multiplication is not real analytic. The authors considered this result as \"surprising\" for the used concept of infinite-dimensional real analyticity for maps $E\\to F$ defined by the property that locally a holomorphic extension $E_{\\mathbb C}\\to F_{\\mathbb C}$ exist. In this note we show that this type of real analyticity is quite rare for composition maps ${\\rm f\\,}\\varphi:x\\mapsto\\varphi\\circ x$ when $\\varphi$ is real analytic. Specifically, we show that the smooth Fr\\'echet space map ${\\rm f\\,}\\varphi:C\\,(\\mathbb R)\\to C\\,(\\mathbb R)$ for real analytic $\\varphi:\\mathbb R\\to\\mathbb R$ is real analytic in the above sense only if $\\varphi$ is the restriction to $\\mathbb R$ of some entire function $\\mathbb C\\to\\mathbb C$. We also discuss the possibility of proving that the set of these \"admissible\" functions $\\varphi$ be \"small\" in the space $A\\,(\\mathbb R)$ of real analytic functions either in the Baire categorical sense, or in the measure theoretic sense of shyness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-13T13:19:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46T20",
      "46T25",
      "46G20",
      "46G05",
      "46G12"
    ],
    "keywords": [
      "composition",
      "smooth frechet space map",
      "real analytic lie group",
      "infinite-dimensional real analyticity",
      "measure theoretic sense"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}