We usually think of ℝn as a set that is equipped with the operations of vector addition and scalar multiplication, defined as follows: if α ∈ ℝ and x, y ∈ ℝn with x = (x1 … xn) and y = (y1, …, yn) then x + y = (x1 + y1, …, xn + yn) and αx = (αx1, …, αxn). These operations endow ℝn with the structure of a vector space as follows.

Definition (Vector space over ℝ) A vector space over ℝ is a triple (X, +, ·), where V is a nonempty set; (x, y) ↦ x + y with x, y ∈ X and (α, x) ↦ α · x with α ∈ ℝ and x ∈ X are two operations called vector addition and scalar multiplication, respectively.