L2 norm

Euclidian Norm

$$\Vert x{\Vert }_2=\sqrt{\sum _{i=1}^n{x}_i^2}$$

The euclidian norm satisfies:

• definiteness: $\Vert x{\Vert }_2=0⟺x=0$
• absolute homogeneity: $\Vert \alpha x{\Vert }_2=|\alpha |\Vert x{\Vert }_2$ for any scalar $\alpha \in \mathbb{R}$
• triangle inequality: $\Vert x+y{\Vert }_2\le \Vert x{\Vert }_2+\Vert y{\Vert }_2$ for any vectors $x,y\in {\mathbb{R}}^n$

The equality $\Vert u+v{\Vert }_2=\Vert u{\Vert }_2+\Vert v{\Vert }_2⟺\exists \alpha \ge 0:u=\alpha v$ holds.

The euclidian norm is derived from the inner product as $\Vert x{\Vert }_2=\sqrt{⟨x,x⟩}$