Editing Series (mathematics) (section)

=== Abelian topological groups ===
Let <math>a : I \to X</math> be a map, also denoted by <math>\left(a_i\right)_{i \in I},</math> from some non-empty set <math>I</math> into a [[Hausdorff space|Hausdorff]] [[Abelian group|abelian]] [[topological group]] <math>X.</math> 
Let <math>\operatorname{Finite}(I)</math> be the collection of all [[Finite set|finite]] [[subset]]s of <math>I,</math> with <math>\operatorname{Finite}(I)</math> viewed as a [[directed set]], [[Partially ordered set|ordered]] under [[Inclusion (mathematics)|inclusion]] <math>\,\subseteq\,</math> with [[Union (set theory)|union]] as [[Join (mathematics)|join]]. 
The family <math>\left(a_i\right)_{i \in I},</math> is said to be {{em|[[unconditionally summable]]}} if the following [[Limit of a net|limit]], which is denoted by <math>\textstyle \sum_{i\in I} a_i</math> and is called the {{em|sum}} of <math>\left(a_i\right)_{i \in I},</math> exists in <math>X:</math>

<math display=block>\sum_{i\in I} a_i := \lim_{A \in \operatorname{Finite}(I)} \ \sum_{i\in A} a_i = \lim \biggl\{\sum_{i\in A} a_i \,: A \subseteq I, A \text{ finite }\biggr\}</math>
Saying that the sum <math>\textstyle S := \sum_{i\in I} a_i</math> is the limit of finite partial sums means that for every neighborhood <math>V</math> of the origin in <math>X,</math> there exists a finite subset <math>A_0</math> of <math>I</math> such that

<math display=block>S - \sum_{i \in A} a_i \in V \qquad \text{ for every finite superset} \; A \supseteq A_0.</math>

Because <math>\operatorname{Finite}(I)</math> is not [[Total order|totally ordered]], this is not a [[limit of a sequence]] of partial sums, but rather of a [[Net (mathematics)|net]].<ref name="Bourbaki">{{cite book|title=General Topology: Chapters 1–4|first=Nicolas|last=Bourbaki|author-link=Nicolas Bourbaki|year=1998|publisher=Springer|isbn=978-3-540-64241-1|pages=261–270}}</ref><ref name="Choquet">{{cite book|title=Topology|first=Gustave|last=Choquet|author-link=Gustave Choquet|year=1966|publisher=Academic Press|isbn=978-0-12-173450-3|pages=216–231}}</ref>

For every neighborhood <math>W</math> of the origin in <math>X,</math> there is a smaller neighborhood <math>V</math> such that <math>V - V \subseteq W.</math>  It follows that the finite partial sums of an unconditionally summable family <math>\left(a_i\right)_{i \in I},</math> form a {{em|[[Cauchy net]]}}, that is, for every neighborhood <math>W</math> of the origin in <math>X,</math> there exists a finite subset <math>A_0</math> of <math>I</math> such that

<math display=block>\sum_{i \in A_1} a_i - \sum_{i \in A_2} a_i \in W \qquad \text{ for all finite supersets } \; A_1, A_2 \supseteq A_0,</math>
which implies that <math>a_i \in W</math> for every <math>i \in I \setminus A_0</math> (by taking <math>A_1 := A_0 \cup \{i\}</math> and <math>A_2 := A_0</math>).

When <math>X</math> is [[Complete topological group|complete]], a family <math>\left(a_i\right)_{i \in I}</math> is unconditionally summable in <math>X</math> if and only if the finite sums satisfy the latter Cauchy net condition. When <math>X</math> is complete and <math>\left(a_i\right)_{i \in I},</math> is unconditionally summable in <math>X,</math> then for every subset <math>J \subseteq I,</math> the corresponding subfamily <math>\left(a_j\right)_{j \in J},</math> is also unconditionally summable in <math>X.</math>

When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group <math>X = \R.</math>

If a family <math>\left(a_i\right)_{i \in I}</math> in <math>X</math> is unconditionally summable then for every neighborhood <math>W</math> of the origin in <math>X,</math> there is a finite subset <math>A_0 \subseteq I</math> such that <math>a_i \in W</math> for every index <math>i</math> not in <math>A_0.</math>  If <math>X</math> is a [[first-countable space]] then it follows that the set of <math>i \in I</math> such that <math>a_i \neq 0</math> is countable.  This need not be true in a general abelian topological group (see examples below).