Euclidean norm - Imported Knowlpedia Content

A Euclidean norm on a Euclidean space $(V,\langle\cdot,\cdot\rangle)$ is the norm defined by

\|x\|=\sqrt{\langle x,x\rangle}\quad\text{for }x\in V.

Remarks

It is the norm induced by an inner product, and it is the standard notion of length in finite-dimensional inner product geometry. On $\mathbb{R}^n$ this is the usual “ $\ell^2$ -length.”

Examples

If $x=(3,4)\in\mathbb{R}^2$ , then $\|x\|=\sqrt{3^2+4^2}=5$ .
If $x=(1,1,1)\in\mathbb{R}^3$ , then $\|x\|=\sqrt{1^2+1^2+1^2}=\sqrt{3}$ .