Assess students' ability to apply the spectral theorem for compact self-adjoint operators on a Hilbert space to solve Fredholm integral equations of the second kind: require them to verify compactness and self-adjointness of symmetric L^2 kernels, derive orthonormal eigenfunction expansions of the integral operator, construct series solutions to (I - λK)φ = f with convergence justification, and use the Fredholm alternative to treat parameter values where 1/λ is an eigenvalue, providing explicit solution or nonexistence examples for given kernels and right-hand sides.