# Norm (mathematics)

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In mathematics, the norm of a vector is its length. For the real numbers the only norm is the absolute value. For spaces with more dimensions the norm can be any function ${\displaystyle p}$ with

1. Scales for real numbers ${\displaystyle a}$, that is ${\displaystyle p(ax)=|a|p(x)}$
2. Function of sum is less than sum of functions, that is ${\displaystyle p(x+y)\leq p(x)+p(y)}$ or the triangle inequality
3. ${\displaystyle p(x)=0}$ if and only if ${\displaystyle x=0}$.

## Examples

1. The one-norm is the sum of absolute values: ${\displaystyle \|x\|_{1}=|x_{1}|+|x_{2}|+...+|x_{N}|.}$ This is like finding the distance from one place on a grid to another by summing together the distances in all directions the grid goes; see Manhattan Distance
2. Euclidean norm is the sum of the squares of the values: ${\displaystyle \|x\|_{2}={\sqrt {x_{1}^{2}+x_{2}^{2}+...+x_{N}^{2}}}}$
3. Maximum norm is the maximum absolute value: ${\displaystyle \|x\|_{\infty }=\max(|x_{1}|,|x_{2}|,...,|x_{N}|)}$