= Solution
Part (i) and the <Cauchy-Schwarz inequality> show
$$
\langle f,g\rangle=\lim_{N\to\infty}\langle S_Nf,g\rangle.
$$
Direct integration of this finite <Fourier partial sum> gives
$$
\langle S_Nf,g\rangle
=\sum_{|n|\leq N}\widehat f(n)\overline{\widehat g(n)}.
$$
Moreover <Bessel's inequality> puts both coefficient sequences in $\ell^2$, so their product series is absolutely convergent by the <Cauchy-Schwarz inequality>. Consequently the cross form of <Parseval's identity> is
$$
\boxed{\frac1{2\pi}\int_{-\pi}^{\pi}f(t)\overline{g(t)}\,dt
=\sum_{n\in\mathbb Z}\widehat f(n)\overline{\widehat g(n)}.}
$$
Taking $g=f$ also gives equality of the squared function norm and the squared coefficient norm.
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