Continuous Functions and Hilbert Spaces - Vision with Direction - page 58

4.5 Schwartz Inequality for Functions, Angles

The Schwartz inequality in Sect. 3.7 was defined for conventional matrices and vec-
tors. An analogue of this inequality for function spaces yields the following result:

Theorem 4.1 (Schwartz inequality II ). The Schwartz inequality

| f, g | ≤ f g (4.7)

where

= f ∗ g f, g (4.8)

holds for functions.

Exercise 4.1. Prove that the Schwartz inequality holds even for functions.

Dividing both sides of inequality (4.7) and subsequently removing the magnitude
operator yields

| f, g | f, g

≤

1 ⇔− 1 ≤

≤

1 (4.9)

f g f g

Accordingly, we can define an “angle” ϕ between the functions f and g as follows:

cos( ϕ )=

f, g

(4.10)

f g

because the right-hand side of the equation varies continuously between − 1 and 1 as

f/ f changes. Notice that cos ϕ =1exactly when f/ f equals g/ g , whereas

it equals − 1 when f/ f equals − g/ g .