---
tag: 759C4C6NXZ
kind: definition
name: Summation in topological groups
msc: [40J05, 22A05]
defines: TXS2H001R4
uses: [082GHNPJ53, 0ZCY0Z80B6, 4DVJQJV0DG, 0S1FJ8VEX9, 0YBDSJARWN]
sources:
  - url: https://en.wikipedia.org/wiki/Series_(mathematics)
---

## Definition

Let $G$ be a topological group whose group operation $+$ is [[commutative|082GHNPJ53]].
Let $I$ be a set, and let $(x_i)_{i \in I}$ be a family of elements of $G$.
For a finite subset $F \subseteq I$ write $s_F = \sum_{i \in F} x_i$.
The family is *summable* with *sum* $s \in G$ if the following holds.
For 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 $F_0 \subseteq F \subseteq I$.
Then one writes $\sum_{i \in I} x_i = s$.

## Remarks

Equivalently, the [[net|4DVJQJV0DG]] $(s_F)$, indexed by the finite subsets of $I$ ordered by inclusion, converges to $s$.
If $G$ is [[Hausdorff|0S1FJ8VEX9]], a family has at most one sum; in Mathlib this is `HasSum.unique`.
The definition does not use any order on $I$, so a bijection $\sigma \colon I \to I$ does not change summability or the sum.

Example: in $\R$, the family $(2^{-n})_{n \ge 0}$ has sum $2$.
For $U \supseteq (2 - \varepsilon, 2 + \varepsilon)$ choose $N$ with $2^{-N} < \varepsilon$ and $F_0 = \set{0, \dots, N}$.
For finite $F \supseteq F_0$ one has $2 - 2^{-N} \le s_F < 2$, so $s_F \in U$.

Non-example: in $\R$, the family $((-1)^{n+1}/n)_{n \ge 1}$ is not summable.
For 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.

In $\R$ 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.