While touching on Fourier series in a PDEs course, our professor basically waved her hands at the concept that $ \int_0^a\sum_{n=1}^{\infty}A_n\sin\frac{n\pi}ax\sin\frac{m\pi}ax\,dx=\sum_{n=1}^{\infty}A_n\int_0^a\sin\frac{n\pi}ax\sin\frac{m\pi}ax\,dx=\frac a2A_m, $ claiming that an arcane knowledge of real analysis was required to prove this true for certain cases.
That explanation left me completely unsatisfied, as I then began to wonder about the kind of functions for which this is true or false.
I have some knowledge of real analysis, and a reference to this proof or a sketch of it would do wonders at resolving this mental craving that I have had since.
Edit 1: I added integration boundaries.