# 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 $1$): From Wikimedia by Quartl

Proofs: 1

Proofs: 1 2 3 4
Propositions: 5 6 7

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983