# Multi-index notation

$\def\a{\alpha}$ $\def\b{\beta}$

An abbreviated form of notation in analysis, imitating the vector notation by single letters rather than by listing all vector components.

## Monomials

A point with coordinates $(x_1,\dots,x_n)$ in the $n$-dimensional space (real, complex or over any other field) is denoted by $x$. For a *multiindex* $\a=(\a_1,\dots,\a_n)\in\Z_+^n$ the expression $x^\a$ denotes the product, $x_\a=x_1^{\a_1}\cdots x_n^{\a_n}$. Other expressions related to multiindices are expanded as follows:
$$
\begin{aligned}
|\a|&=\a_1+\cdots+\a_n\in\Z_+^n,
\\
\a!&=\a_1!\cdots\a_n!\qquad\text{(as usual,}0!=1!=1),
\\
x^\a&=x_1^{\a_1}\cdots x_n^{\a_n},
\\
\a\pm\b&=(\a_1\pm\b_1,\dots,\a_n\pm\b_n)\in\Z^n,
\end{aligned}
$$

**How to Cite This Entry:**

Multi-index notation.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Multi-index_notation&oldid=25750