Question

8. 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

Inner product for real functions: $\langle f,g\rangle = \int\limits_{-\infty}^{\infty }f(x)g(x)dx$ .

If $f(x)$ and $g(x)$ are normalized, then $\langle f,f\rangle = \int\limits_{-\infty}^{\infty }f(x)^2dx=1$ and $\langle g,g\rangle = \int\limits_{-\infty}^{\infty }g(x)^2dx=1$ .

We need to prove, that $\langle f+g, f-g\rangle =0$ .

$\langle f+g,f-g\rangle = \int\limits_{-\infty}^{\infty }\big(f(x)+g(x)\big)\cdot \big( f(x)-g(x)\big)dx= \int\limits_{-\infty}^{\infty }\big(f(x)^2-g(x)^2\big)dx= \int\limits_{-\infty}^{\infty }f(x)^2dx-\int\limits_{-\infty}^{\infty }g(x)^2dx=1-1=0.$

So, $f(x)+g(x)$ and $f(x)-g(x)$ are orthogonal