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 $\operatorname{Supp} \left({f}\right)$ be finite.

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

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


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