= Solution
Insert the inverse <Fourier transforms> of $f$ and its complex conjugate, initially under enough decay to justify interchanging <integrals>:
$$
\begin{aligned}
\int_{\mathbb R}|f(x)|^2\,dx
&=\frac1{(2\pi)^2}\int_{\mathbb R}\int_{\mathbb R}\widetilde f(k)\overline{\widetilde f(\ell)}\left[\int_{\mathbb R}e^{i(k-\ell)x}\,dx\right]dk\,d\ell\\
&=\frac1{2\pi}\int_{\mathbb R}\int_{\mathbb R}\widetilde f(k)\overline{\widetilde f(\ell)}\delta(k-\ell)\,dk\,d\ell\\
&=\boxed{\frac1{2\pi}\int_{\mathbb R}|\widetilde f(k)|^2\,dk}.
\end{aligned}
$$
The middle step uses the stated <Dirac delta distribution> identity, and the last uses its sampling property. This is the <Parseval identity> for the chosen normalization. Approximation extends the result to <square-integrable functions>, including the function used below.
Back to article page
