Definition:Multiindex

Definition 1
Let $\displaystyle m = \prod_{j \mathop \in J} X_j^{k_j}$ be a mononomial in the indexed set $\left\{{X_j: j \in J}\right\}$.

Such a mononomial can be expressed implicitly and more compactly by referring only to the sequence of indices:
 * $k = \left \langle {k_j}\right \rangle_{j \in J}$

and write $m = \mathbf X^k$ without explicit reference to the indexing set.

Such an expression is called a multiindex (or multi-index).

Definition 2
Let $J$ be a set.

A $J$-multiindex is a sequence of natural numbers indexed by $J$:
 * $\displaystyle k = \left \langle {k_j}\right \rangle_{j \in J}$

with only finitely many of the $k_j$ non-zero.

Definition 3
A multiindex is an element of $\Z^J$, the free $\Z$-module on $J$, an abelian group of rank over $\Z$ equal to the cardinality of $J$.

Also known as
Some sources hyphenate for clarity: multi-index.

Also see

 * Addition of Multiindices

Linguistic note
The plural is multiindices (or multi-indices).