MATH 540

Overview

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.

Learning Outcomes

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.