{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$.\nA *Fréchet algebra* over $\\mathbb{K}$ is an associative algebra $A$ over $\\mathbb{K}$ with a topology such that:\n\n1. $A$ with this topology is a [[Fréchet space|3GAA614673]], that is, a complete metrizable locally convex space;\n2. the multiplication $A \\times A \\to A$, $(a, b) \\mapsto ab$, is jointly continuous.\n\nLet $(p_n)_{n \\ge 1}$ be an increasing sequence of [[seminorms|FP5GVCN8GV]] defining the topology.\nThen condition 2 holds if and only if for each $n$ there are $m \\ge n$ and $C_n \u003e 0$ with $p_n(ab) \\le C_n \\, p_m(a) \\, p_m(b)$ for all $a, b \\in A$.\n\n## Remarks\n\nThe algebra is called *locally multiplicatively convex* (or m-convex) if the topology is given by seminorms with $p_n(ab) \\le p_n(a) p_n(b)$.\nSome authors include this condition in the definition.\nEvery Banach algebra is a Fréchet algebra, with the single seminorm given by the norm.\n\nExample: $A = C(\\R)$, the continuous complex functions on $\\R$, with $p_n(f) = \\max_{\\abs{x} \\le n} \\abs{f(x)}$.\nEach $p_n$ is a seminorm with $p_n(fg) \\le p_n(f) p_n(g)$, and the sequence is increasing.\nConvergence for all $p_n$ is uniform convergence on compact sets, and a locally uniform limit of continuous functions is continuous, so $A$ is complete.\nSo $C(\\R)$ is an m-convex Fréchet algebra.\n\nNon-example: the polynomials $\\C[x]$ with the same seminorms satisfy every condition except completeness in clause 1.\nThe Taylor polynomials of $e^x$ form a Cauchy sequence for every $p_n$, but their limit $e^x$ is not a polynomial.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "AW4T3NQYVF",
    "kind": "definition",
    "msc": [
      "46H05",
      "46A04"
    ],
    "name": "Fréchet algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fr%C3%A9chet_algebra"
      }
    ],
    "tag": "JTJ8VQA1JC",
    "uses": [
      "3GAA614673",
      "FP5GVCN8GV"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/JTJ8VQA1JC/",
  "links": [
    {
      "field": "defines",
      "target": "AW4T3NQYVF"
    },
    {
      "field": "uses",
      "target": "3GAA614673"
    },
    {
      "field": "uses",
      "target": "FP5GVCN8GV"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/JTJ8VQA1JC.md",
  "msc": [
    "46H05",
    "46A04"
  ],
  "name": "Fréchet algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JTJ/JTJ8VQA1JC.md",
  "status": null,
  "tag": "JTJ8VQA1JC",
  "uri": "m2://JTJ8VQA1JC",
  "url": "https://math2.tamnd.com/wiki/frechet-algebra-definition/",
  "writing": {
    "tag": "JTJ8VQA1JC",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}