Question

If the real normalized functions f(x) and g(x) are not orthogonal, show that their sum f(x) +g(x) and their difference f(x)−g(x) are orthogonal. 

Answer 1

If two functions are orthogonal, then their scalar product is equal to zero. Let's check:

$$\int_\R(f(x)+g(x))(f(x)-g(x))dx = \int_\R f^2(x) -g^2(x)dx=\\ =\int_\R f^2(x)dx -\int_\R g^2(x)dx$$

As far as f(x) and g(x) are normalized,

$$\int_\R f^2(x)dx=\int_\R g^2(x)dx = 1$$

Thus, obtain:

$$\int_\R(f(x)+g(x))(f(x)-g(x))dx = 1-1=0$$

Hence, functions f(x) +g(x) and f(x)−g(x) are orthogonal.