= Solution
For the <weighted inner product>, the factor $w$ cancels the $w^{-1}$ in the <Sturm-Liouville operator>. Assuming the coefficients are real, two applications of <integration by parts> give
$$
\begin{aligned}
\langle f,Lg\rangle_w-\langle Lf,g\rangle_w
&=\int_a^b[-f^*(pg')'+(pf'^*)'g]dx\\
&=\left[p(f'^*g-f^*g')\right]_a^b.
\end{aligned}
$$
This is the boundary form in the <Lagrange identity for a Sturm-Liouville operator>. The potential terms cancel; if $q$ were complex, an additional integral of $(q-q^*)f^*g$ would remain, so suitable real coefficients are part of formal self-adjointness. The endpoint restrictions in part (a) eliminate the displayed form for every pair of functions in the domain.
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