Definition:P-Product Metric/General Definition

Definition
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 $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.

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

The $p$-product metric on $\mathcal A$ is defined as:
 * $\displaystyle d_p \left({x, y}\right) = \left({\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^p}\right)^{\frac 1 p}$

where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$.

The metric space:
 * $\displaystyle \mathcal M_p := \left({\prod_{i \mathop = 1}^n A_{i'}, d_p}\right)$

can be called the $p$-product of $M_{1'}, M_{2'}, \ldots, M_{n'}$.

Real Vector Space
This metric is often found in the context of a real vector space $\R^n$:

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

Also see

 * $p$-Product Metric is Metric

Note
Note that while $d_1, d_2, \ldots, d_\infty$ are all topologically equivalent, this is not the case with $d_0$.

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.