In mathematics, the product metric is a definition of metric on the Cartesian product of two metric spaces. As described below, the p product metric of the Cartesian product of n metric spaces is the p norm of the n-vector of the norms of the n subspaces:
:d_p(\mathbf{x}_1,\dots,\mathbf{x}_n) = \|(d_1(\mathbf{x}_1), \dots, d_n(\mathbf{x}_n))\|_p
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 \times Y by
:d_{p} \left( (x_{1}, y_{1}) , (x_{2}, y_{2}) \right) := \left( d_{X} (x_{1}, x_{2})^{p} + d_{Y} (y_{1}, y...