Plancherel theorem
= Plancherel theorem
{c}
{title2=$\|\widehat f\|_2=(2\pi)^{n/2}\|f\|_2$}
{wiki}
For the unnormalized angular-frequency transform,
$$
\int_{\mathbb R^n}|\widehat f(\xi)|^2\,d\xi
=(2\pi)^n\int_{\mathbb R^n}|f(x)|^2\,dx.
$$
After normalization, the Fourier transform extends unitarily to $L^2$. For integrable $f$ and $\widehat f$, inversion and Fubini give the identity directly.