inner product
(i) ⟨(au + bv), w⟩ = a⟨u, w⟩ + b⟨v, w⟩.
(ii) ⟨u, v⟩ = ⟨v, u⟩.
(iii) ∥v∥2 ≥ 0 and if ⟨v, v⟩ = 0, then v = 0.
The norm (or length) ∥v∥ of a vector then equals 
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
(ii)′ 
to ensure that ⟨v, v⟩ is real. See also Hilbert space.
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