{
  "id": "1606.07977",
  "version": "v1",
  "published": "2016-06-25T23:48:20.000Z",
  "updated": "2016-06-25T23:48:20.000Z",
  "title": "Subspace Condition for Bernstein Lethargy Theorem",
  "authors": [
    "Asuman GÜven Aksoy",
    "Monairah Al-Ansari",
    "Caleb Case",
    "Qidi Peng"
  ],
  "comment": "9 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we consider a condition on subspaces in order to improve bounds given in the Bernstein's Lethargy Theorem (BLT) for Banach spaces. Let $d_1 \\geq d_2 \\geq \\dots d_n \\geq \\dots > 0$ be an infinite sequence of numbers converging to $0$, and let $Y_1 \\subset Y_2 \\subset \\dots\\subset Y_n \\subset \\dots \\subset X$ be a sequence of closed nested subspaces in a Banach space $X$ with the property that $\\overline{Y}_{n}\\subset Y_{n+1}$ for all $n\\ge1$. We prove that there exists $c \\in (0,1]$ and an element $x_c \\in X$ such that $$ c d_n \\leq \\rho(x_c, Y_n) \\leq \\min (4, \\tilde{a}) c\\, d_n. $$ Here, $\\rho(x, Y_n)= \\inf \\{||x-y||: \\,\\,y\\in Y_n\\}$, $$\\tilde{a} =\\sup_{n\\ge1}\\sup_{\\left \\{q_{i} \\right \\}}\\left \\{a_{n_{i+1}-1}^{-3}\\right \\}$$ where the sequence $\\{a_n\\}$ is defined as: for all $ n \\geq 1 $, $$ a_n = \\inf_{l \\geq n} \\, \\inf_{q \\in \\langle q_l, q_{l+1},\\dots \\rangle} \\frac{\\rho(q,Y_l)}{||q||} $$ in which each point $q_n$ is taken from $Y_{n+1} \\setminus Y_{n}$, and satisfies $\\inf\\limits_{n\\ge1} a_n > 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-25T23:48:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A25",
      "41A50",
      "41A65"
    ],
    "keywords": [
      "bernstein lethargy theorem",
      "subspace condition",
      "banach space",
      "bernsteins lethargy theorem",
      "infinite sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}