# Product metric

﻿
Product metric

In mathematics, the product metric is a definition of metric on the Cartesian product of two metric spaces.

Definition

Let $\left(X, d_\left\{X\right\}\right)$ and $\left(Y, d_\left\{Y\right\}\right)$ be metric spaces and let $1 leq p leq + infty$. Define the $p$-product metric $d_\left\{p\right\}$ on $X imes Y$ by

:$d_\left\{p\right\} left\left( \left(x_\left\{1\right\}, y_\left\{1\right\}\right) , \left(x_\left\{2\right\}, y_\left\{2\right\}\right) ight\right) := left\left( d_\left\{X\right\} \left(x_\left\{1\right\}, x_\left\{2\right\}\right)^\left\{p\right\} + d_\left\{Y\right\} \left(y_\left\{1\right\}, y_\left\{2\right\}\right)^\left\{p\right\} ight\right)^\left\{1/p\right\}$ for $1 leq p < infty;$

:$d_\left\{infty\right\} left\left( \left(x_\left\{1\right\}, y_\left\{1\right\}\right) , \left(x_\left\{2\right\}, y_\left\{2\right\}\right) ight\right) := max left\left\{ d_\left\{X\right\} \left(x_\left\{1\right\}, x_\left\{2\right\}\right), d_\left\{Y\right\} \left(y_\left\{1\right\}, y_\left\{2\right\}\right) ight\right\}.$

for $x_\left\{1\right\}, x_\left\{2\right\} in X$, $y_\left\{1\right\}, y_\left\{2\right\} in Y$.

Wikimedia Foundation. 2010.