Product metric

In mathematics, a product metric is a metric on the Cartesian product of finitely many metric spaces ${\displaystyle (X_{1},d_{X_{1}}),\ldots ,(X_{n},d_{X_{n}})}$which metrizes the product topology. The most prominent product metrics are the p product metrics for a fixed ${\displaystyle p\in [0,\infty ]}$ : It is defined as the p norm of the n-vector of the distances measured in n subspaces:

${\displaystyle d_{p}((x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n}))=\|\left(d_{X_{1}}(x_{1},y_{1}),\ldots ,d_{X_{n}}(x_{n},y_{n})\right)\|_{p}}$

For $p=\infty$ this metric is also called the sup metric:

${\displaystyle d_{\infty }((x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n})):=\max \left\{d_{X_{1}}(x_{1},y_{1}),\ldots ,d_{X_{n}}(x_{n},y_{n})\right\}.}$

Choice of norm

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 product (in the category theory sense) uses the sup metric.

References

• Deza, Michel Marie; Deza, Elena (2009), Encyclopedia of Distances, Springer-Verlag, p. 83.