{
  "id": "1605.04592",
  "version": "v1",
  "published": "2016-05-15T18:50:10.000Z",
  "updated": "2016-05-15T18:50:10.000Z",
  "title": "On a Theorem of S. N. Bernstein for Banach Spaces",
  "authors": [
    "Asuman G. Aksoy",
    "Qidi Peng"
  ],
  "comment": "16 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We prove that if $X$ is an infinite-dimensional Banach space and $\\{Y_n\\}$ is a nested sequence of subspaces of $ X$ such that $ Y_n \\subset Y_{n+1}$ and $\\overline{Y_n} \\subset Y_{n+1}$ for any $ n \\in \\mathbb{N} $, and if $\\{d_n \\}$ be a decreasing sequence of positive numbers tending to 0, then for any $0< c\\leq 1$ there exists $ x_c \\in X$ such that the distance $\\rho(x_c, Y_n)$ from $x_c$ to $Y_n$ satisfies $$ c d_n \\leq \\rho(x_c,Y_n) \\leq 4c d_n. $$ We prove the above by first improving Borodin's result \\cite{Borodin} for Banach spaces by weakening the condition on the sequence $\\{d_n\\}$. Lastly, we compare subsequences $\\{d_{\\varphi(n)}\\}$ under different choices of $\\varphi(n)$ and examine their effects on approximation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-15T18:50:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A25",
      "41A50",
      "41A65",
      "46B20"
    ],
    "keywords": [
      "infinite-dimensional banach space",
      "first improving borodins result",
      "paper contains",
      "decreasing sequence",
      "approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}