# Definition:Multiindex

## Definition

### 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 $J$.

### Modulus

Let $k = \left \langle {k_j}\right \rangle_{j \mathop \in J}$ be a multiindex.

The modulus of such a multiindex $k$ is defined by:

$\displaystyle \left|{k}\right| = \sum_{j \mathop \in J} k_j$

## Also known as

Some sources hyphenate for clarity: multi-index.

## Linguistic note

The plural is multiindices (or multi-indices).