{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "4KX81Q85DT"
    },
    {
      "field": "uses",
      "target": "87FM669CDD"
    },
    {
      "field": "uses",
      "target": "JTJ8VQA1JC"
    },
    {
      "field": "uses",
      "target": "NPR76WM5DQ"
    },
    {
      "field": "uses",
      "target": "YBAPZPEDWQ"
    }
  ],
  "body": "\n## Summary\n\nA seminorm on a real or complex vector space has all the properties of a [[norm|0V2WP9B1MF]], except that it may vanish on nonzero vectors.\nSeminorms are the building blocks of [[locally convex spaces|B6GY2KJMWW]]: a family of seminorms defines a topology, and every locally convex topology arises in this way.\nMany function spaces are described by seminorms rather than by one norm.\nAn example is the space of continuous functions on $\\R$ with uniform convergence on compact sets.\n\n## Explanations\n\n### L1 human\n\nA seminorm measures the size of a vector, but it is allowed to be blind to some directions.\nIt still scales correctly and still obeys the triangle inequality.\nOn $\\R^2$, the function $p(x, y) = \\abs{x}$ is a seminorm: it measures only the first coordinate.\nIt gives $p(0, 5) = 0$ although $(0, 5) \\ne 0$, so it is not a norm.\nOn functions, $p(f) = \\abs{f(0)}$ is a seminorm that only looks at the value at $0$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46A03"
    ],
    "name": "Seminorm",
    "same_as": {
      "wikidata": "Q1416088",
      "wikipedia": "Seminorm"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Seminorm"
      }
    ],
    "specializes": [
      "K3356N0SAX"
    ],
    "tag": "FP5GVCN8GV"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/FP5GVCN8GV/",
  "links": [
    {
      "field": "specializes",
      "target": "K3356N0SAX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/FP5GVCN8GV.md",
  "msc": [
    "46A03"
  ],
  "name": "Seminorm",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FP5/FP5GVCN8GV.md",
  "status": null,
  "tag": "FP5GVCN8GV",
  "uri": "m2://FP5GVCN8GV",
  "url": "https://math2.tamnd.com/wiki/seminorm/",
  "writing": {
    "tag": "FP5GVCN8GV",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}