View source for Cartesian Products of Metric Spaces

In mathematics, a '''product metric''' is a metric on the Cartesian product of finitely many metric spaces <math>(X_1,d_{X_1}),\ldots,(X_n,d_{X_n})</math> which metrizes the product topology. The most prominent product metrics are the ''p'' '''product metrics''' for a fixed <math>p\in[1,\infty)</math> :
It is defined as the ''p'' norm of the ''n''-vector of the distances measured in ''n'' subspaces:
:<math>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</math>

For <math>p=\infty</math> this metric is also called the sup metric:
:<math>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\}.</math>

==Choice of norm==
For Euclidean spaces, using the L<sub>2</sub> 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.

==The case of Riemannian manifolds==

For Riemannian manifolds <math>(M_1,g_1)</math> and <math>(M_2,g_2)</math>, the '''product metric''' <math>g=g_1\oplus g_2</math> on <math>M_1\times M_2</math> is defined by 

:<math>g(X_1+X_2,Y_1+Y_2)=g_1(X_1,Y_1)+g_2(X_2,Y_2)</math> 

for <math>X_i,Y_i\in T_{p_i}M_i</math> under the natural identification <math>T_{(p_1,p_2)}(M_1\times M_2)=T_{p_1}M_1\oplus T_{p_2}M_2</math>.

== Licensing == 
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Product_metric Product metric, Wikipedia] under a CC BY-SA license