Question: Prove Parseval Identity for $f \in C(\Bbb T) $ $2\pi$ periodic continuous functions
$$ \frac{1}{2 \pi} \int_{-\pi}^\pi |f(x)|^2 dx =\sum_{n=-\infty}^\infty |\hat f(n)|^2 $$
Thoughts:
We are somehow supposed to use the fact that exists $g\in C^1(\Bbb T)$ s.t. $f$ and $g$ are very close. Tried defining a function $h=f-g$, and then the identity is true for the $g$ part but not for the $f$ part...