= Solution
Set $g=f$. The <cross-correlation> at zero is $h(0)=\int_{\mathbb R}|f(x)|^2dx$, while part (i) gives $\widehat h=|\widehat f|^2$. Evaluating the <inverse Fourier transform> at zero yields the <Parseval identity>
$$
\boxed{\int_{\mathbb R}|f(x)|^2dx=\frac1{2\pi}\int_{\mathbb R}|\widehat f(k)|^2dk.}
$$
This derivation is immediate for <Schwartz functions>, for which all integrations and <Fourier inversion> are justified. Approximation by such functions and completeness then extend the identity to <L2 space>, defining its <Fourier transform> by the corresponding norm limit. This is the <Plancherel theorem> extension; no absolute integrability of an arbitrary square-integrable function is presumed.
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