{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe argue by contradiction: in infinite dimension, Riesz's lemma gives unit vectors that stay far apart.\n\n1. Let $\\bar{B} = \\{x \\in X : \\|x\\| \\le 1\\}$ be compact, and suppose that $X$ is not [[finite-dimensional|018W3V3TCT]]. Then $X$ has no finite spanning set. In particular $X \\neq \\{0\\}$.\n2. We build unit vectors $x_1, x_2, \\ldots$ with $\\|x_m - x_n\\| \\ge \\tfrac{1}{2}$ for $m \\neq n$. Choose any $x_1 \\neq 0$ and replace it by $x_1 / \\|x_1\\|$.\n3. Given $x_1, \\ldots, x_n$, let $Y_n$ be their [[linear span|0HR573CHNM]]. It is a [[linear subspace|0EPTJA5QV4]], and $Y_n \\neq X$ because $X$ has no finite spanning set.\n4. $Y_n$ with the restricted norm is a finite-dimensional normed space. So it is complete, by [[finite-dimensional normed spaces are complete|0ABP2F2ZP7]].\n5. Its metric is the restriction of the metric of $X$. So $Y_n$ is closed in $X$, by [[complete subspaces are closed|0W41XG4W60]].\n6. By [[Riesz's lemma|097C7KCAT5]] with $\\alpha = \\tfrac{1}{2}$, there is $x_{n+1}$ with $\\|x_{n+1}\\| = 1$ and $\\|x_{n+1} - y\\| \\ge \\tfrac{1}{2}$ for all $y \\in Y_n$.\n7. For $m \u003e n$, we have $x_n \\in Y_{m-1}$. So $\\|x_m - x_n\\| \\ge \\tfrac{1}{2}$, by step 6 for $x_m$.\n8. All $x_n$ lie in $\\bar{B}$. As $\\bar{B}$ is compact in the metric space $X$, it is [[sequentially compact|0CARW59WD2]], by [[this result|0K8SGQSV7G]].\n9. So some subsequence $(x_{n_k})$ [[converges|06JYPJ4WHH]] to a point $z$. There is $k_0$ with $\\|x_{n_k} - z\\| \u003c \\tfrac{1}{4}$ for $k \\ge k_0$.\n10. Then $\\|x_{n_{k_0}} - x_{n_{k_0 + 1}}\\| \\le \\|x_{n_{k_0}} - z\\| + \\|z - x_{n_{k_0 + 1}}\\| \u003c \\tfrac{1}{2}$, with $n_{k_0} \\neq n_{k_0 + 1}$. This contradicts step 7.\n11. So $X$ is finite-dimensional.\n\n## Idea\n\nIn infinite dimension, each finite-dimensional subspace is a closed proper subspace. Riesz's lemma then gives a new unit vector far from all previous ones. The resulting sequence has no convergent subsequence, so the unit ball cannot be compact.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "46B50"
    ],
    "name": "Proof that a compact closed unit ball implies finite dimension",
    "proves": "0VZVV8VME5",
    "sources": [
      {
        "locator": "Characterization of finite dimension",
        "url": "https://en.wikipedia.org/wiki/Riesz%27s_lemma"
      }
    ],
    "tag": "G96C77E433",
    "uses": [
      "097C7KCAT5",
      "0K8SGQSV7G",
      "0CARW59WD2",
      "0ABP2F2ZP7",
      "0W41XG4W60",
      "018W3V3TCT",
      "0HR573CHNM",
      "0EPTJA5QV4",
      "06JYPJ4WHH"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/G96C77E433/",
  "links": [
    {
      "field": "proves",
      "target": "0VZVV8VME5"
    },
    {
      "field": "uses",
      "target": "097C7KCAT5"
    },
    {
      "field": "uses",
      "target": "0K8SGQSV7G"
    },
    {
      "field": "uses",
      "target": "0CARW59WD2"
    },
    {
      "field": "uses",
      "target": "0ABP2F2ZP7"
    },
    {
      "field": "uses",
      "target": "0W41XG4W60"
    },
    {
      "field": "uses",
      "target": "018W3V3TCT"
    },
    {
      "field": "uses",
      "target": "0HR573CHNM"
    },
    {
      "field": "uses",
      "target": "0EPTJA5QV4"
    },
    {
      "field": "uses",
      "target": "06JYPJ4WHH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/G96C77E433.md",
  "msc": [
    "46B50"
  ],
  "name": "Proof that a compact closed unit ball implies finite dimension",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/G96/G96C77E433.md",
  "status": null,
  "tag": "G96C77E433",
  "uri": "m2://G96C77E433",
  "url": "https://math2.tamnd.com/wiki/proof-that-a-compact-closed-unit-ball-implies-finite-dimension/",
  "writing": {
    "tag": "G96C77E433",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}