{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $c$ be the set of real [[convergent sequences|0W4CN9E2W5]] $x = (x_n)_{n \\ge 1}$, with the norm $\\norm{x}_\\infty = \\sup_n \\abs{x_n}$.\nThen $c$ is a [[Banach space|06GZ956657]].\n\nCheck.\nEvery convergent sequence is [[bounded|0A8JNBGT2Z]], so $c$ is a linear subspace of the Banach space $\\ell^\\infty$ of bounded sequences.\nIt suffices to show that $c$ is closed in $\\ell^\\infty$.\nLet $x^{(k)} \\in c$ tend to $x$ in $\\ell^\\infty$.\nFor all $k$, $m$ and $n$,\n$$\\abs{x_m - x_n} \\le 2 \\norm{x - x^{(k)}}_\\infty + \\abs{x^{(k)}_m - x^{(k)}_n}.$$\nChoose $k$ to make the first term small, then $m, n$ large to make the second term small.\nSo $x$ is a Cauchy sequence of real numbers, and it converges.\n\n## Remarks\n\nThe sequences with limit $0$ form the closed subspace $c_0$ of codimension $1$ in $c$.\nThe map $x \\mapsto (\\ell, x_1 - \\ell, x_2 - \\ell, \\dots)$, with $\\ell = \\lim_n x_n$, is a linear isomorphism of $c$ onto $c_0$.\nBut $c$ and $c_0$ are not isometric.\nThe constant sequence $(1, 1, \\dots)$ is an extreme point of the unit ball of $c$, while the unit ball of $c_0$ has no extreme points.\nThe dual of $c$ is isometric to $\\ell^1$.\n\n## Explanations\n\n### L1 human\n\nThe space $c$ collects all sequences of real numbers that settle down to a limit.\nTwo sequences are close when they are close at every index.\nA uniform limit of sequences that settle down also settles down, so the space has no holes.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "instance_of": [
      "06GZ956657"
    ],
    "kind": "example",
    "msc": [
      "46B45",
      "46B25"
    ],
    "name": "Space of convergent sequences c",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/C_space"
      }
    ],
    "tag": "2N4RJCW5JK",
    "uses": [
      "06GZ956657",
      "0W4CN9E2W5",
      "0A8JNBGT2Z"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/2N4RJCW5JK/",
  "links": [
    {
      "field": "instance_of",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "0W4CN9E2W5"
    },
    {
      "field": "uses",
      "target": "0A8JNBGT2Z"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/2N4RJCW5JK.md",
  "msc": [
    "46B45",
    "46B25"
  ],
  "name": "Space of convergent sequences c",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2N4/2N4RJCW5JK.md",
  "status": null,
  "tag": "2N4RJCW5JK",
  "uri": "m2://2N4RJCW5JK",
  "url": "https://math2.tamnd.com/wiki/space-of-convergent-sequences-c/",
  "writing": {
    "tag": "2N4RJCW5JK",
    "kind": "example",
    "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
      }
    ]
  }
}