An inner product is a generalization of the scalar product. An inner product on ℝⁿ is usually denoted ⟨u, v⟩ rather than u·v and satisfies:

1. (i) ⟨(au + bv), w⟩ = a⟨u, w⟩ + b⟨v, w⟩.

2. (ii) ⟨u, v⟩ = ⟨v, u⟩.

3. (iii) ∥v∥² ≥ 0 and if ⟨v, v⟩ = 0, then v = 0.

The norm (or length) ∥v∥ of a vector then equals $\sqrt{⟨\mathbf{v},\mathbf{v}⟩}.$ The distance between points v and w is then defined as ∥v–w∥ and the angle between v and w equals

(see Cauchy-Schwarz inequality). A real inner product space is a real vector space with an inner product. A complex inner product space is a complex vector space with an inner product that satisfies instead

1. (ii)′ $⟨\mathbf{u},\mathbf{v}⟩=\overline{⟨\mathbf{v},\mathbf{u}⟩}$

to ensure that ⟨v, v⟩ is real. See also Hilbert space.