Cartesian Products of Metric Spaces

In mathematics, a product metric is a metric on the Cartesian product of finitely many metric spaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 $p\in [1,\infty )$ : It is defined as the p norm of the n-vector of the distances measured in n subspaces:

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

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

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

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.

The case of Riemannian manifolds

For Riemannian manifolds Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M_1,g_1)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M_2,g_2)} , the product metric Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=g_1\oplus g_2} on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_1\times M_2} is defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(X_1+X_2,Y_1+Y_2)=g_1(X_1,Y_1)+g_2(X_2,Y_2)}

for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i,Y_i\in T_{p_i}M_i} under the natural identification Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{(p_1,p_2)}(M_1\times M_2)=T_{p_1}M_1\oplus T_{p_2}M_2} .

Licensing

Content obtained and/or adapted from:

• Product metric, Wikipedia under a CC BY-SA license