{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $V$ and $W$ be normed vector spaces over $\\mathbb{R}$ and let $A \\colon V \\to W$ be a bounded linear map.\nThen $A$ is Fréchet differentiable at each point $x \\in V$, and its [[Fréchet derivative|08EVNYZF42]] at $x$ is $A$ itself: $DA(x) = A$.\n\n## Remarks\n\nThe proof is one line: $A(x + h) - A(x) - A h = 0$ for each $h$, so the remainder is $0$.\nFor $V = \\mathbb{R}^n$ and $W = \\mathbb{R}^m$, each linear map is bounded. So the derivative of $x \\mapsto M x$, for an $m \\times n$ matrix $M$, is the linear map with matrix $M$ at each point.\nAn affine map $x \\mapsto A x + b$ has the same derivative $A$ at each point, because the derivative of a constant map is $0$.\n\n## Explanations\n\n### L1 human\n\nThe best linear approximation of a linear map is the map itself.\nThis is the analogue of the rule that the derivative of $x \\mapsto c x$ is the constant $c$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "26B05"
    ],
    "name": "Derivative of a continuous linear map",
    "role": "proposition",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative"
      }
    ],
    "tag": "0CYYAF8AD2",
    "uses": [
      "08EVNYZF42"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0CYYAF8AD2/",
  "links": [
    {
      "field": "uses",
      "target": "08EVNYZF42"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0CYYAF8AD2.md",
  "msc": [
    "26B05"
  ],
  "name": "Derivative of a continuous linear map",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0CY/0CYYAF8AD2.md",
  "status": null,
  "tag": "0CYYAF8AD2",
  "uri": "m2://0CYYAF8AD2",
  "url": "https://math2.tamnd.com/wiki/derivative-of-a-continuous-linear-map/",
  "writing": {
    "tag": "0CYYAF8AD2",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}