Definition:P-Product Metric

Definition
Let $M_{1'} = \struct {A_{1'}, d_{1'} }$ and $M_{2'} = \struct {A_{2'}, d_{2'} }$ 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:
 * $\map {d_p} {x, y} := \paren {\paren {\map {d_{1'} } {x_1, y_1} }^p + \paren {\map {d_{2'} } {x_2, y_2} }^p}^{1/p}$

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

The metric space $\mathcal M_p := \struct {A_{1'} \times A_{2'}, d_p}$ is the $p$-product (space) of $M_{1'}$ and $M_{2'}$.

Real Number Plane
This metric is often found in the context of a real number plane $\R^2$:

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:

Also known as
The $p$-product metric is also referred to as the $p$-norm, but that term is also used for a slightly more general concept.

Also see

 * $p$-Product Metric is Metric

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 $p$th exponential that defines the metric itself.

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