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 ${\displaystyle x=0}$.

1. ↑ "Norm - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2020-08-24.