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.
$\blacksquare$