Mathematics Course 321

Notes

Some (not all) parts of the course notes will be in the form of a handout or will be available here.

Chapter 1 (Banach Spaces): Normed spaces. Metric spaces (review). Examples of normed spaces. Complete metric spaces (review). Completion of a metric space (review). Baire Category Theorem. Banach spaces (classical examples). Linear operators (examples, inclusions among Lp spaces and norms of the inclusions, isomorphism of n-dimensional spaces). Open mapping, closed graph and uniform boundedness theorems.
Chapter 2 (Fundamentals): Axiom of choice; partial oprders; Zorn's lemma and some applications.
Chapter 3 (Dual Spaces and the Hahn-Banach Theorem): Definition of the dual of a normed space, prooof that it is a Banach space in general and some examples of duals. The proof of the Hahan Banach theorem for real scalars and the case of complex scalars, which relies on the real case. Reflexivity is defined.
Chapter 4 (Hilbert Spaces): Basic properties of Hilbert spaces. Separable if and only if countable orthonormal basis. Fourier series basis. Riesz representation theorem. Parallegrom identity. The operator algebra B(H) and definition of a C*-algebra.
Chapter 5 (Application: Fourier Series): Some basic facts about Fourier series using functional analysis. Riemann-Lebesgue lemma, use of Dirichlet kernel. Outline of proofs that the `converse' of the Riemann-Lebesgue lemma does not hold, and that Fourier series of continuous functions do not necessarily converge pointwise.