{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$.\nA *normed algebra* over $\\mathbb{K}$ is an associative algebra $A$ over $\\mathbb{K}$ with a [[norm|0V2WP9B1MF]] $\\norm{\\cdot}$ on the vector space $A$ such that\n$$\\norm{x y} \\le \\norm{x} \\norm{y} \\quad \\text{for all } x, y \\in A.$$\nIf $A$ has a unit $1$, many authors also ask $\\norm{1} = 1$.\n\n## Remarks\n\nThe inequality makes the product $A \\times A \\to A$ continuous.\nA normed algebra that is complete as a [[normed vector space|03J5RAKB30]] is a Banach algebra.\n\nExample: the algebra $C([0,1])$ of continuous real functions with $\\norm{f}_\\infty = \\sup_t \\abs{f(t)}$.\nFor each $t$, $\\abs{f(t) g(t)} \\le \\norm{f}_\\infty \\norm{g}_\\infty$, so $\\norm{fg}_\\infty \\le \\norm{f}_\\infty \\norm{g}_\\infty$.\nThe polynomial functions form a subalgebra, which is a normed algebra that is not complete.\n\nNon-example: $M_2(\\R)$ with the norm $\\norm{a} = \\max_{i,j} \\abs{a_{ij}}$.\nFor the matrix $a$ with all entries $1$, $a^2$ has all entries $2$, so $\\norm{a^2} = 2 \u003e 1 = \\norm{a}^2$.\nThe operator norm on $M_2(\\R)$, in contrast, does make it a normed algebra.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "4DJ4V5VBSH",
    "kind": "definition",
    "msc": [
      "46H05"
    ],
    "name": "Normed algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Normed_algebra"
      }
    ],
    "tag": "7VS7JZ3Y5B",
    "uses": [
      "03J5RAKB30",
      "0V2WP9B1MF"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/7VS7JZ3Y5B/",
  "links": [
    {
      "field": "defines",
      "target": "4DJ4V5VBSH"
    },
    {
      "field": "uses",
      "target": "03J5RAKB30"
    },
    {
      "field": "uses",
      "target": "0V2WP9B1MF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7VS7JZ3Y5B.md",
  "msc": [
    "46H05"
  ],
  "name": "Normed algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7VS/7VS7JZ3Y5B.md",
  "status": null,
  "tag": "7VS7JZ3Y5B",
  "uri": "m2://7VS7JZ3Y5B",
  "url": "https://math2.tamnd.com/wiki/normed-algebra-definition/",
  "writing": {
    "tag": "7VS7JZ3Y5B",
    "kind": "definition",
    "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
      }
    ]
  }
}