# Definition:P-Product Metric/Real Vector Space

## Definition

Let $\R^n$ be an $n$-dimensional real vector space.

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

The $p$-product metric on $\R^n$ is defined as:

$\ds \map {d_p} {x, y} := \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^p}^{\frac 1 p}$

where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.