section_path: Generalizations/Summations over arbitrary index sets/Abelian topological groups
section_name: Abelian topological groups
paragraph_text: Abelian topological groups
  
  Let a: I → X {\displaystyle a:I\to X} be a map, also denoted by (a i) i ∈ I, {\displaystyle \left(a_{i}\right)_{i\in I},} from some non-empty set I {\displaystyle I} into a Hausdorff abelian topological group X. {\displaystyle X.} Let Finite (I) {\displaystyle \operatorname {Finite} (I)} be the collection of all finite subsets of I, {\displaystyle I,} with Finite (I) {\displaystyle \operatorname {Finite} (I)} viewed as a directed set, ordered under inclusion ⊆ {\displaystyle \,\subseteq \,} with union as join. The family (a i) i ∈ I, {\displaystyle \left(a_{i}\right)_{i\in I},} is said to be unconditionally summable if the following limit, which is denoted by ∑ i ∈ I a i {\displaystyle \sum _{i\in I}a_{i}} and is called the sum of (a i) i ∈ I, {\displaystyle \left(a_{i}\right)_{i\in I},} exists in X: {\displaystyle X:}
  
  ∑ i ∈ I a i:= lim A ∈ Finite (I) ∑ i ∈ A a i = lim { ∑ i ∈ A a i: A ⊆ I, A finite } {\displaystyle \sum _{i\in I}a_{i}:=\lim _{A\in \operatorname {Finite} (I)}\ \sum _{i\in A}a_{i}=\lim \left\{\sum _{i\in A}a_{i}\,:A\subseteq I,A{\text{ finite }}\right\}} Saying that the sum S:= ∑ i ∈ I a i {\displaystyle S:=\sum _{i\in I}a_{i}} is the limit of finite partial sums means that for every neighborhood V {\displaystyle V} of the origin in X, {\displaystyle X,} there exists a finite subset A 0 {\displaystyle A_{0}} of I {\displaystyle I} such that
  
  S − ∑ i ∈ A a i ∈ V for every finite superset A ⊇ A 0. {\displaystyle S-\sum _{i\in A}a_{i}\in V\qquad {\text{ for every finite superset}}\;A\supseteq A_{0}.}
  
  Because Finite (I) {\displaystyle \operatorname {Finite} (I)} is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a net.
  
  For every neighborhood W {\displaystyle W} of the origin in X, {\displaystyle X,} there is a smaller neighborhood V {\displaystyle V} such that V − V ⊆ W. {\displaystyle V-V\subseteq W.} It follows that the finite partial sums of an unconditionally summable family (a i) i ∈ I, {\displaystyle \left(a_{i}\right)_{i\in I},} form a Cauchy net, that is, for every neighborhood W {\displaystyle W} of the origin in X, {\displaystyle X,} there exists a finite subset A 0 {\displaystyle A_{0}} of I {\displaystyle I} such that
  
  ∑ i ∈ A 1 a i − ∑ i ∈ A 2 a i ∈ W for all finite supersets A 1, A 2 ⊇ A 0, {\displaystyle \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},} which implies that a i ∈ W {\displaystyle a_{i}\in W} for every i ∈ I ∖ A 0 {\displaystyle i\in I\setminus A_{0}} (by taking A 1:= A 0 ∪ { i } {\displaystyle A_{1}:=A_{0}\cup \{i\}} and A 2:= A 0 {\displaystyle A_{2}:=A_{0}}).
  
  When X {\displaystyle X} is complete, a family (a i) i ∈ I {\displaystyle \left(a_{i}\right)_{i\in I}} is unconditionally summable in X {\displaystyle X} if and only if the finite sums satisfy the latter Cauchy net condition. When X {\displaystyle X} is complete and (a i) i ∈ I, {\displaystyle \left(a_{i}\right)_{i\in I},} is unconditionally summable in X, {\displaystyle X,} then for every subset J ⊆ I, {\displaystyle J\subseteq I,} the corresponding subfamily (a j) j ∈ J, {\displaystyle \left(a_{j}\right)_{j\in J},} is also unconditionally summable in X. {\displaystyle X.}
  
  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 X = R. {\displaystyle X=\mathbb {R}.}
  
  If a family (a i) i ∈ I {\displaystyle \left(a_{i}\right)_{i\in I}} in X {\displaystyle X} is unconditionally summable then for every neighborhood W {\displaystyle W} of the origin in X, {\displaystyle X,} there is a finite subset A 0 ⊆ I {\displaystyle A_{0}\subseteq I} such that a i ∈ W {\displaystyle a_{i}\in W} for every index i {\displaystyle i} not in A 0. {\displaystyle A_{0}.} If X {\displaystyle X} is a first-countable space then it follows that the set of i ∈ I {\displaystyle i\in I} such that a i ≠ 0 {\displaystyle a_{i}\neq 0} is countable. This need not be true in a general abelian topological group (see examples below).