Product metric


Product metric

In mathematics, the product metric is a definition of metric on the Cartesian product of two metric spaces.

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 imes Y by

:d_{p} left( (x_{1}, y_{1}) , (x_{2}, y_{2}) ight) := left( d_{X} (x_{1}, x_{2})^{p} + d_{Y} (y_{1}, y_{2})^{p} ight)^{1/p} for 1 leq p < infty;

:d_{infty} left( (x_{1}, y_{1}) , (x_{2}, y_{2}) ight) := max left{ d_{X} (x_{1}, x_{2}), d_{Y} (y_{1}, y_{2}) ight}.

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


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