{
  "id": "2307.05536",
  "version": "v1",
  "published": "2023-07-08T15:01:02.000Z",
  "updated": "2023-07-08T15:01:02.000Z",
  "title": "$\\ell^1$-Bounded Sets",
  "authors": [
    "Christopher Heil",
    "Pu-Ting Yu"
  ],
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "A subset $M$ of a separable Hilbert space $H$ is $\\ell^1$-bounded if there exists a Riesz basis $\\mathcal{F} = \\{e_n\\}_{n \\in \\mathbb{N}}$ for $H$ such that $\\sup_{x \\in M} \\sum_{n \\in \\mathbb{N}} |\\langle x, e_n\\rangle| < \\infty.$ A similar definition for $\\ell^1$-frame-bounded sets is made by replacing Riesz bases with frames. This paper derives properties of $\\ell^1$-bounded sets, operations on the collection of $\\ell^1$-bounded sets, and the relation between $\\ell^1$-boundedness and $\\ell^1$-frame-boundedness. Some open problems are stated, several of which have intriguing implications.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-08T15:01:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riesz basis",
      "paper derives properties",
      "open problems",
      "separable hilbert space",
      "replacing riesz bases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}