Real and Complex values.

Question

$f$ is a $2\pi$-periodic function defined on $\mathbb{R}$.I reached this step,$$\int_{-\pi}^{\pi}\overline{f}(\theta)e^{in\theta} d\theta = \int_{-\pi}^{\pi}f(\theta)e^{in\theta} d\theta,$$ Does this means that f is real valued ? If so what is the justification ?

Answer 1

I think this is the answer but I may be wrong.

You can write $f(t)$=$f_1(t)$+i$f_2(t)$.

From the equality you've written you get that $2i\int_{-\pi}^{\pi}f_2(\theta)e^{in\theta} d\theta=0$.

{$e^{in\theta}$} is a hilbert base for the space of $L^2[-\pi,\pi]$, thus if $<e^{in\theta},f>=0$ for every $n \in \Bbb Z$ then $f=0$.

Therefore $f_2(\theta)=0$.