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:


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

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

Also see

 * P-Product Metric on Real Vector Space is Metric


 * P-Product Metrics on Real Vector Space are Topologically Equivalent


 * Definition:Discrete Metric

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