A Proof of Cauchy’s Inequality and a New Generalization

Extract

If a₁, …, a_n and b₁, …, b_n are two sets of real numbers, then (a₁b₁ + … + a_nb_b)² ≤ (a₁² + … + a_n²)(b₁² + … + b_n²), the equality holding if and only if either 3 a real number λ such that a_r — λb_r (r = 1, …, n) with some b_r ≠ 0 or b₁ = … = b_n = 0. Write a = (a₁, …, a_n) and b = (b₁, …, b_n). Suppose a ≠ 0 and b ≠ 0, for otherwise the inequality is trivial. Write с = a — λb.

Choose λ so as to make b, c=0, i.e take

the equality holding if and only if e=0, i.e if and only if a=λb,

2. Suppose x₁...,x_n and y₁ are two sets of linearly independent vectors in the space E defined on the previous note.