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


Special Cases

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

  1. Proposition: Maximum Norm as a Limit of p-Norms

Proofs: 1 2 3 4
Propositions: 5 6 7

Thank you to the contributors under CC BY-SA 4.0!




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