Students should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor. For more detailed information visit the Math 540 Wiki page.
Normed spaces: Basics, Banach spaces, Special linear operators, Duality, Adjoints of bounded linear operators, Second duals, Weak and weak-star topologies, Banach-Alaoglu theorem, Finite-dimensional spaces, Baire category theorem, Hahn-Banach extension theorem, Banach-Steinhaus theorem, Open mapping theorem, Closed graph theorem, Bounded inverse theorem
Inner product spaces: Basics, Structure, and important theorems.
Spectral theory: Banach algebras, Bounded operators on Banach spaces, Compact operators on Banach spaces.