Assess graduate-level understanding of compact operators on Hilbert spaces: precise definitions and examples, spectral properties (point spectrum, eigenvalue accumulation at 0, multiplicity, spectral theorem for compact self-adjoint operators), Fredholm alternative, approximation by finite-rank operators, and the singular value decomposition (existence of orthonormal singular vectors and relation of singular values to spectrum). Include conceptual questions and short computations (e.g., integral operators, diagonal compact operators).