= Solution
For sufficiently rapidly decaying functions, substitute the <inverse Fourier transform> of $g$ and interchange the <integrals>:
$$
\begin{aligned}
\int_{-\infty}^{\infty}f(x)^*g(x)\,dx&=\frac1{2\pi}\int_{-\infty}^{\infty}\widetilde g(k)\left[\int_{-\infty}^{\infty}f(x)^*e^{ikx}\,dx\right]dk\\
&=\boxed{\frac1{2\pi}\int_{-\infty}^{\infty}\widetilde f(k)^*\widetilde g(k)\,dk.}
\end{aligned}
$$
The bracketed <integral> is the <complex conjugate> of $\widetilde f(k)$. This is <Parseval's identity> for the chosen normalization. One may first prove it for <Schwartz functions>, where the interchanges are justified, and then extend it to $L^2$ by the <Plancherel theorem>.
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