{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0C13E623WD"
    },
    {
      "field": "uses",
      "target": "0ZTEH055YF"
    }
  ],
  "body": "\n## Statement\n\nLet $V$ and $W$ be normed spaces and let $T \\colon V \\to W$ be linear.\nThen $T$ is [[continuous|0S3SMRH0Z5]] if and only if $T$ is bounded, that is, there is a real $C \\ge 0$ with $\\|Tx\\|_W \\le C \\|x\\|_V$ for all $x \\in V$.\n\n## Remarks\n\nIf $T$ is bounded, then $\\|Tx - Ty\\|_W = \\|T(x - y)\\|_W \\le C \\|x - y\\|_V$. Thus $T$ is [[Lipschitz|0TMZ21WKCN]], and so [[uniformly continuous|07DCKDXRBP]].\nConversely, if $T$ is continuous at $0$, there is $\\delta \u003e 0$ with $\\|Tx\\|_W \\le 1$ when $\\|x\\|_V \\le \\delta$. By linearity, $\\|Tx\\|_W \\le \\|x\\|_V / \\delta$ for all $x$.\nThus, for a linear map, continuity at one point already gives continuity everywhere.\nThe name bounded means that $T$ maps bounded sets to bounded sets. It does not mean that $T(V)$ is bounded.\n\n## Explanations\n\n### L1 human\n\nFor linear maps, continuity is the same as a simple estimate. The length of $Tx$ is at most a fixed multiple of the length of $x$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "47B01"
    ],
    "name": "Bounded linear maps are continuous",
    "role": "theorem",
    "sources": [
      {
        "locator": "Equivalence of boundedness and continuity",
        "url": "https://en.wikipedia.org/wiki/Bounded_operator"
      }
    ],
    "tag": "0F3K4C7XYW",
    "uses": [
      "0S3SMRH0Z5",
      "0TMZ21WKCN",
      "07DCKDXRBP"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0F3K4C7XYW/",
  "links": [
    {
      "field": "uses",
      "target": "0S3SMRH0Z5"
    },
    {
      "field": "uses",
      "target": "0TMZ21WKCN"
    },
    {
      "field": "uses",
      "target": "07DCKDXRBP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0F3K4C7XYW.md",
  "msc": [
    "47B01"
  ],
  "name": "Bounded linear maps are continuous",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0F3/0F3K4C7XYW.md",
  "status": null,
  "tag": "0F3K4C7XYW",
  "uri": "m2://0F3K4C7XYW",
  "url": "https://math2.tamnd.com/wiki/bounded-linear-maps-are-continuous/",
  "writing": {
    "tag": "0F3K4C7XYW",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}