This seems to be a stupid question, but I just can't figure it out. I'm giving a proof that given every sequence $\{a_n\}$ of $\ell^2$ there is a function $f\in L^2[-\pi,\pi]$ with Fourier coefficients given precisely as the $a_n$.
Of course, the obvious choice is $f(x) = \sum_{n=-\infty}^\infty a_n e^{inx}$. I have quickly verified that $f\in L^2[-\pi,\pi]$. The only task remaining is to verify by hand that $f$ has Fourier coefficients precisely $a_n.$ I wanted to write: \begin{align} \hat{f}(m) &= \frac{1}{2\pi} \int_{-\pi}^\pi \left(\sum_{n=-\infty}^\infty a_n e^{inx} \right)e^{-imx} dx\\ &= \sum_{n=-\infty}^\infty a_n \frac{1}{2\pi}\int_{-\pi}^\pi e^{i(n-m)x} dx\\ &= a_m \end{align} but I'm not sure whether I can interchange the summation and the integral; the most likely method is DCT, but I'm not seeing it. Would appreciate all help.