If a 1, …, a n and b 1, …, b n are two sets of real numbers, then (a 1 b 1 + … + a n b b)2 ≤ (a 1 2 + … + a n 2)(b 1 2 + … + b n 2), the equality holding if and only if either 3 a real number λ such that ar — λbr (r = 1, …, n) with some br ≠ 0 or b 1 = … = b n = 0. Write a = (a 1, …, an ) and b = (b 1, …, bn ). 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...,xn and y1 are two sets of linearly independent vectors in the space E defined on the previous note.