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 $\map {\operatorname{Supp} } f$ be finite.

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

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


 * $\ds \sum_{s \mathop \in S} \map f s = \sum_{s \mathop \in \map {\operatorname {Supp} } f} \map g s$