= Solution
First, setting $f=g=y_n$ in the <self-adjoint operator> identity shows that $\lambda_n\langle y_n,y_n\rangle_w$ is real, so every <eigenvalue> is real. For two <eigenfunctions>,
$$
\lambda_n\langle y_m,y_n\rangle_w
=\langle y_m,\mathcal Ly_n\rangle_w
=\langle\mathcal Ly_m,y_n\rangle_w
=\lambda_m\langle y_m,y_n\rangle_w.
$$
Distinct <eigenvalues> therefore imply
$$
\boxed{\langle y_m,y_n\rangle_w=\int_a^b y_m^*y_nw\,dx=0.}
$$
This is <orthogonality of Sturm-Liouville eigenfunctions>.
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