Euclidean Space

For $k\in\mathbb{N}$, Euclidean space $\mathbb{R}^k$ is the set of all $k$-tuples $(x_1,\dots,x_k)$ with $x_i\in\mathbb{R}$, where $\mathbb{R}$ denotes the real numbers . Equivalently, $\mathbb{R}^k$ is the $k$-fold Cartesian product of $\mathbb{R}$ with itself.

With coordinatewise addition and scalar multiplication by $\mathbb{R}$, $\mathbb{R}^k$ is a vector space . It carries a standard inner product and the associated Euclidean norm .

Examples:

• $\mathbb{R}^1$ is (canonically) identified with the real line.
• $\mathbb{R}^2$ models the Euclidean plane; points and vectors can both be represented as ordered pairs.
• $\mathbb{R}^3$ models ordinary 3-dimensional space with coordinates $(x,y,z)$.