Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm

A generalization of the Euclidean Norm is defined by the p-norm. Let $p$ be a real number $\ge 1$. Let $x=(x_1,x_2,\ldots x_n)$ be a vector of a vector space $V$ over the field of real numbers $\mathbb R$ or the field of complex numbers $\mathbb C$.

The p-norm of $x$ is defined by

$$||x||_p:=\left(\sum_{\nu=1}^n|x_\nu|^p\right)^{1/p}.$$

Special Cases

$p=1$: taxicab norm $$||x||_1:=\sum_{\nu=1}^n|x_\nu|.$$
$p=2$: Euclidean Norm $$||x||_2:=\sqrt{\sum_{\nu=1}^n|x_\nu|^2}.$$
$p=3$: 3-Norm $$||x||_3:=\sqrt[ 3 ]{\sum_{\nu=1}^n|x_\nu|^3}.$$
...
$p\to\infty$: maximum norm $$||x||_\infty:=\max_{\nu} |x_\nu|.$$

The p-norm can be visualized for two dimensional vectors $x=(x_1,x_2)$ as follows: The below figure shows some unity circles (i.e. the values of coordinates $(x_1,x_2)$, for which the p-norm $||x||_p$ takes the value $1$):

vector-p-norms