Test graduate students' understanding of the spectral theorem for compact self-adjoint operators on Hilbert spaces: state the theorem and hypotheses; explain why eigenvalues are real, nonzero spectrum is discrete with finite multiplicities and 0 is the only possible accumulation point; prove existence of an orthonormal eigenbasis and the resulting spectral (diagonal) decomposition; outline key proof techniques (Riesz–Schauder theory, variational/Rayleigh characterization) and apply these ideas to compute the spectrum and eigen-expansion of a concrete compact integral operator example.