In mathematics, the product metric is a definition of metric on the Cartesian product of two metric spaces in the category of metric spaces. Another family of choices of metric on the product space is the p product metric, described below, of the Cartesian product of n metric spaces is the p norm of the n-vector of the norms of the n subspaces:
Note that this differs, in general; see (Choice of norm) section below.
Let and be metric spaces and let . Define the -product metric on by
for and .
Choice of norm
For Euclidean spaces, using the L2 norm gives rise to the Euclidean metric in the product space; however, any other choice of p will lead to a topologically equivalent metric space. In the category of metric spaces (with Lipschitz maps having Lipschitz constant 1), the sup norm is used.