Chapter 10.3, Problem 33E

Let $\left\{f_n\left(x\right)\right\}$ be an orthogonal set of functions on the interval $\left[a,b\right]$ with respect to the weight function $w\left(x\right)$. Show that they satisfy the Pythagorean property

${\Vert f_m+f_n\Vert }^2={\Vert f_m\Vert }^2+{\Vert f_n\Vert }^2$

if $m\ne n$.