Prove that the set of functions
$C$ = { $f_{0}(x)= \sqrt{\frac{1}{\pi}}$, $f_{n}(x)= \sqrt{\frac{2}{\pi}}\cdot \cos(nx), n \in \Bbb N$ }
is an orthonormal basis in $L^{2}(0, \pi)$.
Hint: Use the facts that the set of continuous functions on compact interval are dense in $L^2$ and every polynomial on compact interval is dense in the set of continuous functions (Stone-Weierstrass theorem). You need to show that $\overline{Sp(C)}$ =$L^2(0, \pi)$.
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