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Math 2.0

Functional analysis Example

p-metric on R^n

Example

Let $n \ge 1$ be an integer and let $p \ge 1$ be a real number. For $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$ , let $d_p(x, y) = \left( \sum_{i=1}^{n} |x_i - y_i|^p \right)^{1/p}.$ Then $d_p$ is a metric on $\mathbb{R}^n$ .

Remarks

The triangle inequality for $d_p$ is the Minkowski inequality. For $p = 1$ it is the taxicab metric, and for $p = 2$ it is the Euclidean metric. For $0 < p < 1$ , the same formula does not give a metric when $n \ge 2$ : the triangle inequality fails. The metric $d_p$ comes from the p-norm: $d_p(x, y) = \|x - y\|_p$ .

Where this leads

19 pages build on this one.

Further reading

Web https://en.wikipedia.org/wiki/Lp_space, The p-norm in finite dimensions

About this page

Tag	`0TFDZN6AXP`
URI	`m2://0TFDZN6AXP`
Aliases	`lp-metric`
Kind	example
Writing	W0 stub
MSC	46B20
Instance of	Metric space

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p-norm on R^n