Product metric

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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:

d_p(\mathbf{x}_1,\dots,\mathbf{x}_n) = \|(d_1(\mathbf{x}_1), \dots, d_n(\mathbf{x}_n))\|_p

Note that this differs, in general; see (Choice of norm) section below.

Definition[edit]

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_{2})^{p} \right)^{1/p} for 1 \leq p < \infty;
d_{\infty} \left( (x_{1}, y_{1}) , (x_{2}, y_{2}) \right) := \max \left\{ d_{X} (x_{1}, x_{2}), d_{Y} (y_{1}, y_{2}) \right\}.

for x_{1}, x_{2} \in X and y_{1}, y_{2} \in Y.

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 sup norm is used.

References[edit]