Norm (mathematics)

In mathematics, the norm of a vector is its length. A vector is a mathematical object that has a size, called the magnitude, and a direction. 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 the following three properties:^[1]

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, ${\displaystyle p(x+y)\leq p(x)+p(y)}$ (also known as the triangle inequality).
3. ${\displaystyle p(x)=0}$ if and only if ${\displaystyle x=0}$.

For a vector ${\displaystyle x}$, the associated norm is written as ${\displaystyle ||x||_{p}}$,^[2] or L${\displaystyle p}$ where ${\displaystyle p}$ is some value. The value of the norm of ${\displaystyle x}$ with some length ${\displaystyle N}$ is as follows:^[3]

${\displaystyle ||x||_{p}={\sqrt[{p}]{|x_{1}|^{p}+|x_{2}|^{p}+...+|x_{N}|^{p}}}}$

The most common usage of this is the Euclidean norm, also called the standard distance formula.

1. The one-norm is the sum of absolute values: ${\displaystyle \|x\|_{1}=|x_{1}|+|x_{2}|+...+|x_{N}|.}$^[2] 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 (also called L2-norm) is the sum of the squares of the values:^[3] ${\displaystyle \|x\|_{2}={\sqrt {x_{1}^{2}+x_{2}^{2}+...+x_{N}^{2}}}}$
3. Maximum norm is the maximum absolute value: ${\displaystyle \|x\|_{\infty }=\max(|x_{1}|,|x_{2}|,...,|x_{N}|)}$
4. When applied to matrices, the Euclidean norm is referred to as the Frobenius norm.
5. L0 norm is the number of non-zero elements present in a vector.

• Magnitude (mathematics)

1. ↑ "Norm - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2020-08-24.
2. ↑ ^{2.0 2.1} "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-24.
3. ↑ ^{3.0 3.1} Weisstein, Eric W. "Vector Norm". mathworld.wolfram.com. Retrieved 2020-08-24.

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