Assess graduate students' ability to apply the spectral theorem for compact self-adjoint operators on Hilbert spaces to solve Fredholm integral equations of the second kind. Tasks: show that a symmetric square-integrable kernel defines a compact self-adjoint integral operator, derive its eigenfunction expansion, express and justify the solution of (I − λK)f = g as a series in eigenfunctions, analyze existence and uniqueness for generic λ and the resonance case when 1/λ is an eigenvalue, and verify convergence in the appropriate Hilbert-space norm; include one concrete kernel for explicit computation.