section_path: Generalizations/Summations over arbitrary index sets/Series in topological vector spaces
section_name: Series in topological vector spaces
paragraph_text: Series in topological vector spaces
  
  If X {\displaystyle X} is a topological vector space (TVS) and (x i) i ∈ I {\displaystyle \left(x_{i}\right)_{i\in I}} is a (possibly uncountable) family in X {\displaystyle X} then this family is summable if the limit lim A ∈ Finite (I) x A {\displaystyle \lim _{A\in \operatorname {Finite} (I)}x_{A}} of the net (x A) A ∈ Finite (I) {\displaystyle \left(x_{A}\right)_{A\in \operatorname {Finite} (I)}} exists in X, {\displaystyle X,} where Finite (I) {\displaystyle \operatorname {Finite} (I)} is the directed set of all finite subsets of I {\displaystyle I} directed by inclusion ⊆ {\displaystyle \,\subseteq \,} and x A:= ∑ i ∈ A x i. {\textstyle x_{A}:=\sum _{i\in A}x_{i}.}
  
  It is called absolutely summable if in addition, for every continuous seminorm p {\displaystyle p} on X, {\displaystyle X,} the family (p (x i)) i ∈ I {\displaystyle \left(p\left(x_{i}\right)\right)_{i\in I}} is summable.If X {\displaystyle X} is a normable space and if (x i) i ∈ I {\displaystyle \left(x_{i}\right)_{i\in I}} is an absolutely summable family in X, {\displaystyle X,} then necessarily all but a countable collection of x i {\displaystyle x_{i}} ’s are zero. Hence, in normed spaces, it is usually only ever necessary to consider series with countably many terms.
  
  Summable families play an important role in the theory of nuclear spaces.