Product metric

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

[edit] Definition

Let (X,dX) and (Y,dY) be metric spaces and let 1 \leq p \leq + \infty. Define the p-product metric dp 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, y_{1}, y_{2} \in Y.