Product metric
In mathematics, a product metric is a metric on the Cartesian product of finitely many metric spaces formula_1 which metrizes the product topology. The most prominent product metrics are the "p" product metrics for a fixed formula_2 : It is defined as the "p" norm of the "n"-vector of the distances measured in "n" subspaces: For formula_4 this metric is also called the sup metric: Choice of norm. For Euclidean spaces, using the L 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. The case of Riemannian manifolds. For Riemannian manifolds formula_6 and formula_7, the product metric formula_8 on formula_9 is defined by for formula_11 under the natural identification formula_12.