norma euclidea

Definizione in Rn

Definiamo per ogni $\underline{x}=(x_1,\dots ,x_N)\in {\mathbb{R}}^N$ la norma euclidea

$$\Vert \underline{x}\Vert =\sqrt{⟨\underline{x},\underline{x}⟩}=\sqrt{x_1^2+\dots +x_N^2}$$

Legame tra norma e distanza euclidea

$\Vert \underline{x}\Vert =d(\underline{x},\underline{0})$ e $d(\underline{x},\underline{y})=\Vert \underline{x}-\underline{y}\Vert$

Proprietà della norma

$i)$ $\Vert \underline{x}\Vert =0\iff \underline{x}=\underline{0}$
$ii)$ $\Vert \lambda \underline{x}\Vert =\left|\lambda \right|\Vert \underline{x}\Vert \forall \lambda \in \mathbb{R}$
$iii)$ $\Vert \underline{x}+\underline{y}\Vert \le \Vert \underline{x}\Vert +\Vert \underline{y}\Vert$

Risorse