{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "3D6XH41A3B"
    },
    {
      "field": "defines",
      "target": "65HSHAH2Y4"
    },
    {
      "field": "uses",
      "target": "8XDYNK6TV0"
    },
    {
      "field": "uses",
      "target": "KFYW0WN9AD"
    }
  ],
  "body": "\n## Summary\n\nA normed field is a [[field|04C5ZBG1WT]] with an absolute value, see the [[definition|65HSHAH2Y4]].\nAn absolute value is a size function that is zero only at $0$, is multiplicative and satisfies the triangle inequality.\nThe size gives a distance, so a normed field is a [[metric space|0VKD1TP7F3]] and a [[topological field|D7YQW9RG24]].\nThe real and complex numbers with the usual [[absolute value|0FFXKTKKQC]] and the rationals with a [[p-adic absolute value|WGN82TCEF2]] are the basic examples.\nNormed fields are the scalars of [[normed vector spaces|03J5RAKB30]] in general functional analysis.\n\n## Explanations\n\n### L1 human\n\nThe absolute value of real numbers tells how big a number is, and $\\abs{x - y}$ tells how far apart two numbers are.\nA normed field is any field with such a size function that behaves well with products and sums.\nThe size of a product is the product of the sizes, and the size of a sum is at most the sum of the sizes.\nFor example, on $\\Q$ one may call a number small when it has many factors of $3$: $\\abs{9}_3 = 1/9$ and $\\abs{1/3}_3 = 3$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "12J05",
      "12J10"
    ],
    "name": "Normed field",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Absolute_value_(algebra)"
      }
    ],
    "specializes": [
      "D7YQW9RG24"
    ],
    "tag": "ZFW9Q3WTAQ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/ZFW9Q3WTAQ/",
  "links": [
    {
      "field": "specializes",
      "target": "D7YQW9RG24"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/ZFW9Q3WTAQ.md",
  "msc": [
    "12J05",
    "12J10"
  ],
  "name": "Normed field",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZFW/ZFW9Q3WTAQ.md",
  "status": null,
  "tag": "ZFW9Q3WTAQ",
  "uri": "m2://ZFW9Q3WTAQ",
  "url": "https://math2.tamnd.com/wiki/normed-field/",
  "writing": {
    "tag": "ZFW9Q3WTAQ",
    "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
      }
    ]
  }
}