{
  "id": "1801.01327",
  "version": "v1",
  "published": "2018-01-04T12:37:47.000Z",
  "updated": "2018-01-04T12:37:47.000Z",
  "title": "Frobenius Theorem in Banach Space",
  "authors": [
    "Jipu Ma"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1407.5198",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\Lambda$ be an open set in Banach space $E$, $M(x)$ for $x\\in \\Lambda$ a subspace in $E$, and $\\mathcal F=\\{M(x)\\}_{x\\in\\Lambda}$. In this paper, we introduce the concept of the co-final set $J(x_0,E_*)$ for $\\mathcal F$ at $x_0\\in \\Lambda$, then prove Frobenius theorem in Banach space where especially, $\\mathrm{dim}\\, M(x)$ maybe $\\infty$. Using the Frobenius theorem, we prove a theorem in advanced calulus that presents several $c^1$ integrable family $\\mathcal F$ at a point with $\\mathrm{dim}\\, M(x) =\\infty$ and trivial co-final set. Let $\\Lambda=B(E,F)\\setminus\\{0\\},\\ M(X)=\\{T\\in B(E,F) : TN(X)\\subset R(X)\\}$, and $\\mathcal F=\\{M(X)\\}_{X\\in\\Lambda}$. The following theorem is proved using Frobenius theorem: if $A\\in\\Lambda$ is double splitting, then $\\mathcal F$ at $A$ is smooth integrable where $J(A,E_*)$ is non-trivial in general. Specially we have the global result as follows, let $\\Phi$ be any one of $F_k(k<\\infty)$ and $\\Phi_{m,n}(\\min\\{m,n\\}<\\infty)$, then $\\Phi$ is smooth submanifold in $B(E,F)$ and tangent to $M(X)$ at any $X$ in $\\Phi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-04T12:37:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46T99",
      "37C05",
      "53C40",
      "58A03"
    ],
    "keywords": [
      "frobenius theorem",
      "banach space",
      "trivial co-final set",
      "open set",
      "smooth submanifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}