Proof for Fourier transform in $L^2$

Question

This question makes me really confused:

Let $f$ and $g$ two functions in $L^2$. Show that: $$\int \widehat f\cdot gdx= \int f\cdot\widehat gdx,$$ where $\widehat f$ is the Fourier transform for the function $f$.

Notice that I need to prove it in $L^2$. Can anyone please help me?

Thanks,

Answer 1

In this thread, it's shown when $f$ and $g$ are $L^1$ functions. Then $f_n:=f\chi_{\{-n\leq f\leq n\}}$ and $g_n=g\chi_{\{-n\leq g\leq n\}}$ approximate respectively $f$ and $g$ in $L^2$, and are functions of $L^1$. So $$\int_{\Bbb R}f_n\widehat{g_n}dx=\int_{\Bbb R}\widehat{f_n}g_ndx.$$

Your task is to show that we can take the limit (use Plancherel's formula).