P-Product Metric on Real Vector Space is Metric

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

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

Let $d_p: \R^n \times \R^n \to \R$ be the $p$-product metric on $\R^n$:


 * $\displaystyle d_p \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left \vert {x_i - y_i} \right \vert^p}\right)^{\frac 1 p}$

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

Then $d_p$ is a metric.

Comment on notation
It can be shown that:
 * $\displaystyle d_\infty \left({x, y}\right) = \lim_{p \to \infty} d_p \left({x, y}\right)$

That is:
 * $\displaystyle \lim_{p \to \infty} \left({\sum_{i \mathop = 1}^n \left \vert {x_i - y_i} \right \vert^p}\right)^{\frac 1 p} = \max_{i \mathop = 1}^n \left\{{\left \vert {x_i - y_i} \right \vert}\right\}$

Hence the notation.