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Product metric - Wikipedia, the free encyclopedia

Product metric

From Wikipedia, the free encyclopedia

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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 $(X, d X)$ 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 $x_{1}, x_{2} \in X$ , $y_{1}, y_{2} \in Y$ .

Retrieved from "http://en.wikipedia.org../../../p/r/o/Product_metric.html"

Category: Metric geometry

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