Chapter 13.3, Problem 88E

Cauchy–Schwarz Inequality The definition u · v = |u| |v| cos θ implies that |u · v| ≤ | u| |v| (because | cos θ| ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and has many consequences.

88. Algebra inequality Show that

${(u_1+u_2+u_3)}^2\le 3(u_1^2+u_2^2+u_3^2),$

for any real numbers u₁, u₂, and u₃. (Hint: Use the Cauchy–Schwarz Inequality in three dimensions with u = 〈u₁, u₂, u₃〉 and choose v in the right way.)