Euclidean Space

Standard Inner product

$$<x,y>:=\sum _{i=1}^n{x}_i{y}_i$$

Norm

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

Distance

The canonical metric on ${\mathbb{R}}^n$ is the Euclidean distance

$$d(x,y):=\Vert x-y{\Vert }_2=\sqrt{\sum _{i=1}^n(x_i-y_i{)}^2}$$

we can check the axioms of a metric (see Axioms)