The norm of a vector v is written <math>\left \| v \right \|</math>
The norm of a vector v is defined by: <MATH>\left \| v \right \| = \sqrt{\left \langle v,v \right \rangle}</MATH>
where:
In Euclidean space, the inner product is the Linear Algebra - Vector Vector Operations.
<math> \begin{array}{crl} v & = &[v_1, v_2, \dots , v_n] \\ \left \| v \right \| & = & \left \| [v_1, v_2, . . . , v_n] \right \| \\ \left \| v \right \| & = & \sqrt{ {v_1}^2 + {v_2}^2 + \dots + {v_n}^2}\\ \left \| v \right \| & = & \sqrt{ \sum v^2_i }\\ \end{array} </math>
For a 2-vector:
<math>
\begin{array}{crl}
u & = & [u_1, u_2] \\
\left \| u \right \| & = & \sqrt{ {u_1}^2 + {u_2}^2 }\\
(\left \| u \right \|)^2 & = & {u_1}^2 + {u_2}^2 \\
\end{array}
</math>
as the Pythagorean theorem, the norm is then the geometric length of its arrow.
Since it plays the role of length, it should satisfy the following norm properties: