{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "TRB69DC8EW"
    }
  ],
  "body": "\n## Summary\n\nA tame Fréchet space is a graded [[Fréchet space|3GAA614673]] of a special kind.\nUp to tame linear isomorphism, it is a direct summand of a space of exponentially decreasing sequences in a Banach space.\n\nThe notion is due to Richard Hamilton.\nTame spaces carry smoothing operators, and this is what the [[Nash–Moser theorem|9ZCCXT3KKP]] needs to run Newton's method with loss of derivatives.\nThe basic example is the space of smooth functions on a compact manifold, graded by the $C^n$ norms.\n\n## Explanations\n\n### L1 human\n\nA space of smooth functions is measured by a whole ladder of norms: the size of the function, of its first derivatives, of its second derivatives, and so on.\nMany natural operators, such as differential operators, cost a fixed number of rungs of this ladder.\nA tame Fréchet space is one in which the ladder behaves like the ladder of weights $e^{nk}$ on a sequence $(f_k)$.\nThen one can smooth an element, trading rungs in a controlled way, and this makes hard implicit function theorems possible.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "46A61",
      "58C15"
    ],
    "name": "Tame Fréchet space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Nash%E2%80%93Moser_theorem"
      }
    ],
    "specializes": [
      "3GAA614673"
    ],
    "tag": "7WPVZZ4YFC"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/7WPVZZ4YFC/",
  "links": [
    {
      "field": "specializes",
      "target": "3GAA614673"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/7WPVZZ4YFC.md",
  "msc": [
    "46A61",
    "58C15"
  ],
  "name": "Tame Fréchet space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7WP/7WPVZZ4YFC.md",
  "status": null,
  "tag": "7WPVZZ4YFC",
  "uri": "m2://7WPVZZ4YFC",
  "url": "https://tamnd.github.io/math2/wiki/tame-frechet-space/",
  "writing": {
    "tag": "7WPVZZ4YFC",
    "kind": "concept",
    "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
      }
    ]
  }
}