{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ be an infinite-dimensional [[Banach space|0BDTTM24DP]].\nLet $(c_n)$ be a sequence of nonnegative real numbers with $\\sum_n c_n^2 \u003c \\infty$.\nThen there is a sequence $(x_n)$ in $X$ with $\\norm{x_n} = c_n$ for all $n$ such that $\\sum_n x_n$ [[converges unconditionally|2ME9M61TJK]].\n\n## Remarks\n\nTaking $c_n = 1/n$ gives a series in $X$ that converges unconditionally but not [[absolutely|0W3K3QPEWT]].\nIn finite dimensions every unconditionally convergent series converges absolutely.\nSo a Banach space is finite-dimensional if and only if unconditional and absolute convergence of series coincide in it.\nIn a Hilbert space, every unconditionally convergent series satisfies $\\sum \\norm{x_n}^2 \u003c \\infty$, so the condition on $(c_n)$ cannot be weakened in general.\nThe proof uses the Dvoretzky–Rogers lemma, a statement about the geometry of finite-dimensional normed spaces.\n\n## Explanations\n\n### L1 human\n\nIn ordinary space, a series that converges in every order also converges when you replace each vector by its length.\nThe theorem says that this fails in every infinite-dimensional Banach space.\nThere are always series whose sum does not depend on the order, although the lengths of the terms add up to infinity.\n\n### L2 human\n\nThe theorem says that absolute and unconditional convergence agree only in finite dimensions.\nIt shows that the example of $\\sum e_n / n$ in $\\ell^2$ is not special to Hilbert space.\nIts proof works inside large finite-dimensional subspaces of $X$, a typical method of local Banach space theory.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "2ME9M61TJK"
    ],
    "kind": "statement",
    "msc": [
      "46B15",
      "40A05"
    ],
    "name": "Dvoretzky–Rogers theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Unconditional_convergence"
      }
    ],
    "tag": "H924SAD12G",
    "uses": [
      "0BDTTM24DP",
      "2ME9M61TJK",
      "0W3K3QPEWT"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/H924SAD12G/",
  "links": [
    {
      "field": "about",
      "target": "2ME9M61TJK"
    },
    {
      "field": "uses",
      "target": "0BDTTM24DP"
    },
    {
      "field": "uses",
      "target": "2ME9M61TJK"
    },
    {
      "field": "uses",
      "target": "0W3K3QPEWT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/H924SAD12G.md",
  "msc": [
    "46B15",
    "40A05"
  ],
  "name": "Dvoretzky–Rogers theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/H92/H924SAD12G.md",
  "status": null,
  "tag": "H924SAD12G",
  "uri": "m2://H924SAD12G",
  "url": "https://math2.tamnd.com/wiki/dvoretzky-rogers-theorem/",
  "writing": {
    "tag": "H924SAD12G",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}