Definition:Summation/Finite Support

Definition
Let $G$ be an abelian group.

Let $S$ be a set.

Let $f : S \to G$ be a mapping.

Let the support $\operatorname{Supp}f$ be finite.

Let $g$ be the restriction of $f$ to $\operatorname{Supp}f$.

The summation of $f$ over $S$, denoted $\displaystyle\sum_{s\mathop\in S} f(s)$, is the summation over the finite set $\operatorname{Supp} f$ of $g$:


 * $\displaystyle\sum_{s\mathop\in S} f(s) = \displaystyle\sum_{s \mathop\in \operatorname{Supp}f} g(s)$

Special cases

 * Definition:Summation over Finite Set, as shown at Summation over Finite Set Equals Summation over Support

Generalizations

 * Definition:Series