Euclidean norm

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

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 $\|x\|=\sqrt{1^2+1^2+1^2}=\sqrt{3}$.