\begin{definition}[Definition:P-Product Metric/General Definition]
Let $M_{1'} = \struct {A_{1'}, d_{1'} }, M_{2'} = \struct {A_{2'}, d_{2'} }, \ldots, M_{n'} = \struct {A_{n'}, d_{n'} }$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.
Let $p \in \R_{\ge 1}$.
The '''$p$-product metric''' on $\AA$ is defined as:
:$\ds \map {d_p} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, y_i} }^p}^{\frac 1 p}$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.
The metric space $\MM_p := \struct {\AA, d_p}$ is the '''$p$-product (space)''' of $M_{1'}, M_{2'}, \ldots, M_{n'}$.
\end{definition}