Parseval pairing on Schwartz space

By [F1], $\overline{g(x)}=\int \overline{\widehat{g}(\xi )}{e}^{-2\pi ix\cdot \xi }d\xi$. The integrand after multiplying by $f(x)$ is jointly measurable and has absolute double integral $\mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }\widehat{g}{\mathrm{\parallel }}_1<\mathrm{\infty }$, by [F1], [F2] and product integration. Hence [F3] gives $\int f\bar{g}=\int \overline{\widehat{g}(\xi )}[\int f(x)e^{-2\pi ix\cdot \xi }dx]d\xi =\int \widehat{f}\bar{\widehat{g}}$. This also proves absolute integrability of the final product; the original product is integrable since $g$ is bounded and $f$ integrable.

Taking $g=f$ gives equality of the nonnegative square integrals, finite by [F2] for the input and by step 1.1 for its transform. Taking nonnegative square roots proves the norm identity. The displayed pairing is linear in its first entry and conjugate-linear in its second directly from integration and conjugation.