Euclidean space ℝ^k

The set of k-tuples of real numbers, viewed as k-dimensional space.

Euclidean space ℝ^k

For an integer $k\ge 1$ , Euclidean space $\mathbb{R}^k$ is the Cartesian product

\mathbb{R}^k := \underbrace{\mathbb{R}\times\cdots\times\mathbb{R}}_{k\ \text{factors}}.

An element $x\in\mathbb{R}^k$ is written $x=(x_1,\dots,x_k)$ with $x_i\in\mathbb{R}$ .

Euclidean spaces are the standard setting for multivariable calculus and for many basic examples in metric space theory. They come equipped with canonical algebraic and geometric structures (addition, scalar multiplication, inner product , norm ).

Examples:

$\mathbb{R}^1=\mathbb{R}$ .
$\mathbb{R}^2=\{(x,y):x,y\in\mathbb{R}\}$ models the plane.
$\mathbb{R}^3=\{(x,y,z):x,y,z\in\mathbb{R}\}$ models ordinary 3D space.