# 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 $\displaystyle{ x = 0 }$.

## Definition

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.

## Examples

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.

## References

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