{
  "id": "1806.07866",
  "version": "v1",
  "published": "2018-06-20T17:50:10.000Z",
  "updated": "2018-06-20T17:50:10.000Z",
  "title": "Angles and Schauder basis in Hilbert spaces",
  "authors": [
    "Bingzhe Hou",
    "Yang Cao",
    "Geng Tian",
    "Xinzhi Zhang"
  ],
  "comment": "6 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\mathcal{H}$ be a complex separable Hilbert space. We prove that if $\\{f_{n}\\}_{n=1}^{\\infty}$ is a Schauder basis of the Hilbert space $\\mathcal{H}$, then the angles between any two vectors in this basis must have a positive lower bound. Furthermore, we investigate that $\\{z^{\\sigma^{-1}(n)}\\}_{n=1}^{\\infty}$ can never be a basis of $L^{2}(\\mathbb{T},\\nu)$, where $\\mathbb{T}$ is the unit circle, $\\nu$ is a finite positive measure which is discrete or absolutely continuous to Lebesgue measure, and $\\sigma: \\mathbb{Z} \\rightarrow \\mathbb{N}$ is an arbitrary surjective and injective map.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-20T17:50:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B15",
      "46B20",
      "47B37"
    ],
    "keywords": [
      "schauder basis",
      "complex separable hilbert space",
      "positive lower bound",
      "unit circle",
      "finite positive measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}