Test graduate students' understanding of weak and strong convergence in Banach and Hilbert spaces: precise definitions and distinctions; characteristic examples and counterexamples (ℓ^p, L^p, weak* behavior); fundamental properties such as uniqueness of limits, boundedness of weakly convergent sequences, lower semicontinuity of the norm, weak sequential compactness and reflexivity; inner-product characterizations in Hilbert spaces; stability under continuous linear operators, Mazur's theorem (convex combinations), and basic applications in variational methods and PDEs.