In the first term of the 2012/2013, I will be giving the course “Advanced Analysis” here at BGU. This is the department’s core functional analysis course for graduate students, though ambitious undergraduate students are also encouraged to take this course, and some of them indeed do. The price to pay is that we do not assume that the students know any functional analysis, and the only formal requisites are a course in complex variables and a course in (point set) topology, as well as a course in measure theory which can be taken concurrently. The price to pay for having no requisites in functional analysis, while still aiming at graduate level course, is that the course is huge: we have five hours of lectures a week. In practice we will actually have six hours of lectures a week, because I will go abroad in the middle of the semester to this conference and workshop in Bangalore. The official syllabus of the course is as follows:
Banach spaces and Hilbert spaces. Basic properties of Hilbert spaces. Topological vector spaces. Banach-Steinhaus theorem; open mapping theorem and closed graph theorem. Hahn-Banach theorem. Duality. Measures on locally compact spaces; the dual of C(X). Weak and weak-* topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. The Stone-Weierstrass theorem. Banach algebras. Spectrum of a Banach algebra element. Gelfand theory of commutative Banach algebras. The spectral theorem for normal operators (in the continuous functional calculus form).
I plan to cover all these topics (with all that is implicitly implied), but I will probably give the whole course a little bend towards my own area of expertise, especially in the exercises and examples. We do have to wait and see who the students are and what their background is before deciding precisely how to proceed. Some notes for the course will appear (in the English language) on this blog. The official course webpage (which is in Hebrew) is behind this link.