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

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.18$