length in a vector space
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 with the following three properties:
- Scales for real numbers , that is, .
- Function of sum is less than sum of functions, that is, (also known as the triangle inequality).
- if and only if .
For a vector , the associated norm is written as , or L where is some value. The value of the norm of with some length is as follows:
The most common usage of this is the Euclidean norm, also called the standard distance formula.
- The one-norm is the sum of absolute values:  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.
- Euclidean norm (also called L2-norm) is the sum of the squares of the values:
- Maximum norm is the maximum absolute value:
- When applied to matrices, the Euclidean norm is referred to as the Frobenius norm.
- L0 norm is the number of non-zero elements present in a vector.
- ↑ "Norm - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2020-08-24.
- ↑ 2.0 2.1 "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-24.
- ↑ 3.0 3.1 Weisstein, Eric W. "Vector Norm". mathworld.wolfram.com. Retrieved 2020-08-24.