Norm

A generalisation of the concept of length, a norm is a function of a class of arithmetic objects, written \(\|x\|\), that returns a real value having the following properties
\[ \begin{align*} \|x\| &\geqslant 0\\ \|ax\| &= |a|\|x\|\\ \|x+y\| &\leqslant \|x\| + \|y\| \end{align*} \]
for arithmetic objects \(x\) and \(y\) and real value \(a\).