For two- and three-dimensional vectors, the fundamental property of dot products,...

For two- and three-dimensional vectors, the fundamental property of dot products,

A · B = |A||B| cos θ, implies that

| A · B | ≤ | A || B | . (2.3.44)

In this exercise, we generalize this to n-dimensional vectors and functions, in which case (2.3.44) is known as Schwarz’s inequality. [The names of Cauchy and Buniakovsky are also associated with (2.3.44).]

(a) Show that |A − γB|² > 0 implies (2.3.44), where γ = A · B/B · B.

(b) Express the inequality using both

*(c) Generalize (2.3.44) to functions. [Hint: Let A · B mean the integral