{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a topological group whose group operation $+$ is [[commutative|082GHNPJ53]].\nLet $I$ be a set, and let $(x_i)_{i \\in I}$ be a family of elements of $G$.\nFor a finite subset $F \\subseteq I$ write $s_F = \\sum_{i \\in F} x_i$.\nThe family is *summable* with *sum* $s \\in G$ if the following holds.\nFor every [[neighbourhood|0ZCY0Z80B6]] $U$ of $s$ there is a finite set $F_0 \\subseteq I$ with $s_F \\in U$ for every finite $F$ with $F_0 \\subseteq F \\subseteq I$.\nThen one writes $\\sum_{i \\in I} x_i = s$.\n\n## Remarks\n\nEquivalently, the [[net|4DVJQJV0DG]] $(s_F)$, indexed by the finite subsets of $I$ ordered by inclusion, converges to $s$.\nIf $G$ is [[Hausdorff|0S1FJ8VEX9]], a family has at most one sum; in Mathlib this is `HasSum.unique`.\nThe definition does not use any order on $I$, so a bijection $\\sigma \\colon I \\to I$ does not change summability or the sum.\n\nExample: in $\\R$, the family $(2^{-n})_{n \\ge 0}$ has sum $2$.\nFor $U \\supseteq (2 - \\varepsilon, 2 + \\varepsilon)$ choose $N$ with $2^{-N} \u003c \\varepsilon$ and $F_0 = \\set{0, \\dots, N}$.\nFor finite $F \\supseteq F_0$ one has $2 - 2^{-N} \\le s_F \u003c 2$, so $s_F \\in U$.\n\nNon-example: in $\\R$, the family $((-1)^{n+1}/n)_{n \\ge 1}$ is not summable.\nFor any finite $F_0$, adding finitely many odd $n$ beyond $F_0$ to $F_0$ makes $s_F$ as large as one likes, because $\\sum_{n \\text{ odd}} 1/n$ diverges.\n\nIn $\\R$ and $\\C$, a family indexed by $\\N$ is summable if and only if it is [[absolutely convergent|0YBDSJARWN]], and then the sum equals the sum of the series.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "TXS2H001R4",
    "kind": "definition",
    "msc": [
      "40J05",
      "22A05"
    ],
    "name": "Summation in topological groups",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Series_(mathematics)"
      }
    ],
    "tag": "759C4C6NXZ",
    "uses": [
      "082GHNPJ53",
      "0ZCY0Z80B6",
      "4DVJQJV0DG",
      "0S1FJ8VEX9",
      "0YBDSJARWN"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/759C4C6NXZ/",
  "links": [
    {
      "field": "defines",
      "target": "TXS2H001R4"
    },
    {
      "field": "uses",
      "target": "082GHNPJ53"
    },
    {
      "field": "uses",
      "target": "0ZCY0Z80B6"
    },
    {
      "field": "uses",
      "target": "4DVJQJV0DG"
    },
    {
      "field": "uses",
      "target": "0S1FJ8VEX9"
    },
    {
      "field": "uses",
      "target": "0YBDSJARWN"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/759C4C6NXZ.md",
  "msc": [
    "40J05",
    "22A05"
  ],
  "name": "Summation in topological groups",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/759/759C4C6NXZ.md",
  "status": null,
  "tag": "759C4C6NXZ",
  "uri": "m2://759C4C6NXZ",
  "url": "https://math2.tamnd.com/wiki/summation-in-topological-groups-definition/",
  "writing": {
    "tag": "759C4C6NXZ",
    "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
      }
    ]
  }
}