Product metric

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, a product metric is a metric on the Cartesian product of finitely many metric spaces which metrizes the product topology. The most prominent product metrics are the p product metrics for a fixed  : It is defined as the p norm of the n-vector of the distances measured in n subspaces:

For this metric is also called the sup metric:

Choice of norm[edit]

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 product (in the category theory sense) uses the sup metric.

References[edit]