{
  "id": "2106.13413",
  "version": "v1",
  "published": "2021-06-25T03:55:47.000Z",
  "updated": "2021-06-25T03:55:47.000Z",
  "title": "Is the free locally convex space $L(X)$ nuclear?",
  "authors": [
    "Arkady Leiderman",
    "Vladimir Uspenskij"
  ],
  "comment": "10 pages",
  "categories": [
    "math.GN",
    "math.FA"
  ],
  "abstract": "Let $X$ be a Tychonoff space containing an infinite compact subset, we observe that the free locally convex space $L(X)$ is not nuclear; moreover, we prove a stronger result: $L(X)$ cannot be embedded in a product of Hilbert spaces. If $X$ is a $k$-space, then $L(X)$ is nuclear if and only if $X$ is countable and discrete. On the other hand, we show that $L(X)$ is nuclear for every projectively countable $P$-space (in particular, for every Lindel\\\"of $P$-space) $X$. In a sharp contrast with the nuclear property, we prove that $L(X)$ can be embedded in a product of reflexive Banach spaces, for every compact space $X$. However, we provide examples of completely metrizable $X$ such that $L(X)$ cannot be embedded in a product of reflexive Banach spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-25T03:55:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A03",
      "46B25",
      "54D30"
    ],
    "keywords": [
      "free locally convex space",
      "reflexive banach spaces",
      "infinite compact subset",
      "stronger result",
      "compact space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}