Cauchy-Schwarz Inequality
The Cauchy-Schwarz inequality is the upper bound of the inner product.
Contents
Description
The inequality is defined as |⟨u, v⟩|2 ≤ ⟨u, u⟩⟨v, v⟩ for two non-zero vectors u and v.
Note that the natural norm for an inner product space is defined as ||a|| = √⟨a, a⟩. By taking the square root of both sides of the Cauchy-Schwarz inequality, the triangle inequality is defined.
The inequality is sometimes proven in the complex plane. This can be a simpler proof and, since C is isomorphic to R2, it is entirely equivalent.