Solution
= Solution
For Schwartz $f,g$, Fourier inversion and Fubini give
$$
\int\widehat f(\xi)\overline{\widehat g(\xi)}\,d\xi
=\int f(x)\overline{g(x)}\,dx.
$$
Taking $g=f$ proves <Plancherel theorem>; polarization gives the full identity.
= Solution
For Schwartz $f,g$, Fourier inversion and Fubini give
$$
\int\widehat f(\xi)\overline{\widehat g(\xi)}\,d\xi
=\int f(x)\overline{g(x)}\,dx.
$$
Taking $g=f$ proves <Plancherel theorem>; polarization gives the full identity.