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.

Let $A_{1'} \times A_{2'}$ be the cartesian product of $A_{1'}$ and $A_{2'}$.

Let $p \in \R_{\ge 1}$.

The $p$-product metric on $A_{1'} \times A_{2'}$ is defined as:
 * $d_p \left({x, y}\right) := \left({\left({d_{1'} \left({x_1, y_1}\right)}\right)^p + \left({d_{2'} \left({x_2, y_2}\right)}\right)^p}\right)^{1/p}$

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

The metric space:
 * $\mathcal M_p := \left({A_{1'} \times A_{2'}, d_p}\right)$

can be called the $p$-product of $M_{1'}$ and $M_{2'}$.

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

Special Cases
Some special cases of the $p$-product metric are:

Relationship with General 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 general Euclidean metrics.

Also see

 * $p$-Product Metrics are Metrics


 * Definition:General Euclidean Metric of which this is a generalization

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.