# Definition:Multiindex

## Contents

## 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**.

## Also see

## Linguistic note

The plural is **multiindices** (or **multi-indices**).