Assess graduate students' mastery of weak and weak-* topologies on Banach spaces: precise definitions, basic properties and examples (e.g., ℓ^p, c0, L^p and their duals), relationships with the norm topology, characterizations of weak and weak-* convergence, and key consequences and theorems (Banach–Alaoglu, Goldstine, reflexivity criteria, separability and sequential compactness distinctions, simple applications such as constructing proofs or counterexamples).