Is $\{\sqrt{\frac{2}{\pi}}\cos{nx}\}, n=1,2,3...$ a complete basis in $L^2([0, \pi])$? The above is an exercise in Beginning Functional Analysis by Saxe. I approach the problem by extending an arbitrary function symmetrically to $-\pi$ and arguing via the convergence of the classical Fourier Series. However, I can’t avoid the need for an initial term $\frac{1}{\sqrt{\pi}}$. I have found this initial term included in one or two places online, but this evidence is tentative.
I am aware of an answered question with regard to the analogous sine series, but to infer that it directly applies is to assume the answer is likewise analogous.

Hint: if $\{e_n\}$ is a complete basis and $\langle x,e_n\rangle=0$ for all $n$ then $x=0$.
