Product metric

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

Definition

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

:$d\_\{p\}\; left(\; (x\_\{1\},\; y\_\{1\})\; ,\; (x\_\{2\},\; y\_\{2\})\; ight)\; :=\; left(\; d\_\{X\}\; (x\_\{1\},\; x\_\{2\})^\{p\}\; +\; d\_\{Y\}\; (y\_\{1\},\; y\_\{2\})^\{p\}\; ight)^\{1/p\}$ for

:$d\_\{infty\}\; left(\; (x\_\{1\},\; y\_\{1\})\; ,\; (x\_\{2\},\; y\_\{2\})\; ight)\; :=\; max\; left\{\; d\_\{X\}\; (x\_\{1\},\; x\_\{2\}),\; d\_\{Y\}\; (y\_\{1\},\; y\_\{2\})\; ight\}.$

for $x\_\{1\},\; x\_\{2\}\; in\; X$, $y\_\{1\},\; y\_\{2\}\; in\; Y$.