Definition:P-Product Metric

Definition
Let $M_{1'} = \left({A_{1'}, d_{1'}}\right)$ and $M_{2'} = \left({A_{2'}, d_{2'}}\right)$ be metric spaces.

Then we may define metrics on the cartesian product $A_{1'} \times A_{2'}$ in the same manner as the generalized Euclidean metric, as follows.

Let $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in A_{1'} \times A_{2'}$.

Let us define the following:


 * $d_1 \left({x, y}\right) = d_{1'} \left({x_1, y_1}\right) + d_{2'} \left({x_2, y_2}\right)$
 * $d_r \left({x, y}\right) = \left({\left({d_{1'} \left({x_1, y_1}\right)}\right)^r + \left({d_{2'} \left({x_2, y_2}\right)}\right)^r}\right)^{\frac 1 r}$
 * $d_\infty \left({x, y}\right) = \max \left\{{d_{1'} \left({x_1, y_1}\right), d_{2'} \left({x_2, y_2}\right)}\right\}$.

Thus $\mathcal M = \left({A_{1'} \times A_{2'}, d_n}\right)$ with $d_n$ as variously defined above.

General Definition
The definition can be extended to the cartesian product of any finite number $n$ of metric spaces.

Let $M_{1'} = \left({A_{1'}, d_{1'}}\right), M_{2'} = \left({A_{2'}, d_{2'}}\right), \ldots, M_{n'} = \left({A_{n'}, d_{n'}}\right)$ be metric spaces.

Let $\displaystyle \mathcal M = \left({\prod_{i \mathop = 1}^n \left({A_{i'}, d_{i'}}\right), d_n}\right)$, where the definition of $d_n$ is defined as:


 * $\displaystyle d_1 \left({x, y}\right) = \sum_{i \mathop = 1}^n d_{i'} \left({x_i, y_i}\right)$
 * $\displaystyle d_r \left({x, y}\right) = \left({\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^r}\right)^{\frac 1 r}$
 * $\displaystyle d_\infty \left({x, y}\right) = \max_{i \mathop = 1}^n \left\{{d_{i'} \left({x_i, y_i}\right)}\right\}$

where $\displaystyle x = \left({x_1, x_2, \ldots, x_n}\right) \in \prod_{i \mathop = 1}^n A_{i'}$ and $\displaystyle y = \left({y_1, y_2, \ldots, y_n}\right) \in \prod_{i \mathop = 1}^n A_{i'}$.

Relationship with Generalized Euclidean Metric
Let each of $\left({A_{i'}, d_{i'}}\right)$ be the real number line $\R$ under the usual metric.

Thus the cartesian product $\displaystyle \prod_{i \mathop = 1}^n \left({A_{i'}, d_{i'}}\right)$ is the $n$-dimensional real vector space $\R^n$.

Then the product space metrics as described here become the generalized Euclidean metrics.

Also see
The product spaces $\mathcal M$ as defined here are metric spaces.

Notation
The notation is awkward, because it is necessary to use a indexing subscript for the $n$ metric spaces contributing to the product, and also for the $r$th exponential that defines the metric itself.

Thus the "prime" notation on the $n$ metric spaces.