Assess graduate students' understanding of weak and strong (norm) convergence in Hilbert spaces: precise definitions, basic properties and implications (strong ⇒ weak; weak + norm convergence criterion), characterization via inner products, examples and counterexamples (orthonormal sequences, projections), and consequences for boundedness, compact operators, and reflexivity. Include short proofs (e.g., uniqueness of weak limits, equivalence conditions), construction or identification of sequences with prescribed convergence behavior, and one applied problem on convergence of operator images. Expect moderate proofs and example construction.