Euclidean Space

math geometry analysis

Definition

Euclidean Space

An $n$-dimensional Euclidean space, denoted ${\mathbb{R}}^n$, is a finite-dimensional vector space over the real numbers equipped with the standard inner product (dot product):

$$⟨\mathbf{x},\mathbf{y}⟩=\sum _{i=1}^n{x}_i{y}_i$$

This inner product induces a norm $\Vert \mathbf{x}\Vert =\sqrt{⟨\mathbf{x},\mathbf{x}⟩}$ and a metric $d(\mathbf{x},\mathbf{y})=\Vert \mathbf{x}-\mathbf{y}\Vert$, making ${\mathbb{R}}^n$ a complete metric space and a Hilbert space.