\begin{definition}[Definition:P-Product Metric]
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 $\MM_p := \struct {A_{1'} \times A_{2'}, d_p}$ is the '''$p$-product (space)''' of $M_{1'}$ and $M_{2'}$.
\end{definition}