Assess graduate students' understanding of the spectral theorem for compact self-adjoint operators on Hilbert spaces and its principal consequences. Items should require stating and proving the theorem, describing the spectrum (discrete nonzero eigenvalues of finite multiplicity accumulating only at 0), proving existence of an orthonormal eigenbasis and the corresponding eigenexpansion, and showing diagonalization by finite-rank approximations. Include applications: integral operators, the Fredholm alternative, variational (Rayleigh quotient) characterizations of eigenvalues, and implications for trace-class and Hilbert–Schmidt operators. Mix short proofs, computations, and brief applied problems to test conceptual and technical mastery.