Prove: Cauchy-Bunjakovski inequality. For any two arbitrary functions f(x) and g(x)
Proof. Let f(x) and g(x) be two functions, not identically equal to zero, given on the interval (a, b). Now choose two arbitrary numbers λ and μ and form the expression
Because the function [λf(x) - μg(x)]2 under the integral sign is nonnegative, we have the following inequality
which, on expansion, is
which can be written as
We now introduce the notation
Using this notation 1) can be written as
3) 2λμC λ2A + μ2B
Note. Using the absolute value for C in 2) is valid because λ and μ are of arbitrary sign.
The inequality 3) is valid for arbitrary values of λ and μ. Consequently we may set
Substituting these values of λ and μ into 3) we get
If we now replace A, B and C by their expressions in 2) we obtain the Cauchy-Bunjakovski inequality.
Source: Mathematics, Its Content, Methods and Meaning. Vol. 3, p. 235