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

Also see

 * Definition:Standard Discrete Metric
 * Definition:Taxicab Metric
 * Definition:Chebyshev Distance


 * $p$-Product Metric on Real Vector Space is Metric


 * $p$-Product Metrics on Real Vector Space are Topologically Equivalent


 * Standard Discrete Metric is not Topologically Equivalent to p-Product Metrics

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