\begin{definition}[Definition:Factorial/Multiindices]
Let $\alpha$ be a multiindex, indexed by a set $J$ such that for each $j \in J$, $\alpha_j \ge 0$.
Then we define:
:$\ds \alpha! = \prod_{j \mathop \in J} \alpha_j!$
where the factorial on the right is a factorial of natural numbers.
Note that by definition, all by finitely many of the $\alpha_j$ are zero, so the product over $J$ is convergent.
\end{definition}