euclidean space

Euclidean Space

Euclidean space is the real coordinate vector space ${\mathbb{R}}^n$ equipped with the standard inner product, the dot product $⟨x,y⟩$.
With the help of the inner product, we can define the euclidean norm as $\Vert x{\Vert }_2=\sqrt{⟨x,x⟩}$.
And with the norm, we can define the distance between two points $x,y\in {\mathbb{R}}^n$ as $d(x,y)=\Vert x-y{\Vert }_2$.
Together, these structures make ${\mathbb{R}}^n$ (also denoted ${\mathbb{E}}^n$) a metric space.