{
  "id": "2010.08192",
  "version": "v1",
  "published": "2020-10-16T06:54:35.000Z",
  "updated": "2020-10-16T06:54:35.000Z",
  "title": "On a question of Pietch",
  "authors": [
    "Yossi Lonke"
  ],
  "comment": "This is a pre-print of an article published in \"Positivity\" (2020). The final authenticated version is available online at: https://doi.org/10.1007/s11117-020-00758-6",
  "doi": "10.1007/s11117-020-00758-6",
  "categories": [
    "math.FA",
    "math.MG"
  ],
  "abstract": "The main result is that a finite dimensional normed space embeds isometrically in $\\ell_p$ if and only if it has a discrete Levy $p$-representation. This provides an alternative answer to a question raised by Pietch, and as a corollary, a simple proof of the fact that unless $p$ is an even integer, the two-dimensional Hilbert space $\\ell_2^2$ is not isometric to a subspace of $\\ell_p$. The situation for $\\ell_q^2$ with $q\\neq 2$ turns out to be much more restrictive. The main result combined with a result of Dor provides a proof of the fact that if $q\\neq 2$ then $\\ell_q^2$ is not isometric to a subspace of $\\ell_p$ unless $q=p$. Further applications concerning restrictions on the degree of smoothness of finite dimensional subspaces of $\\ell_p$ are included as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-16T06:54:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B05",
      "46B45"
    ],
    "keywords": [
      "main result",
      "finite dimensional normed space embeds",
      "finite dimensional subspaces",
      "two-dimensional hilbert space",
      "discrete levy"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}