P-Product Metric on Real Vector Space is Metric/Proof 2

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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 $\R^n$:

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

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


Then $d_p$ is a metric.


Proof

This is an instance of $p$-Product Metric is Metric.

$\blacksquare$