Math 8800: Seminar in analysis

Syllabus:

Textbooks: W. Rudin: Real and Complex analysis
A Zygmund: Measure and Integral
A.A. Kirillov: Problems and theorems in functional analysis
Gelbaum: Problems in analysis

Instructor: A. Magyar, 603B, office hours: Thursday: 1-2pm
Phone: 542-2543, Email: magyar@math.uga.edu

Grades: A-F based on course work and homework problems

Classes: Presentation of homework assignments and problem discussions, Wednesdays 11:30am – 1pm, Room 302

Topics: - Basic topology of Euclidean spaces, continuity
- Differentiability in one and several variables – review of Calculus
- Borel and Lebesgue measure and measurable functions, theorems of
Egorov and Luzin
- Integrable functions, L^p – spaces, convergence theorems
- Sigma algebras, abstract measures, Radon – Nikodym theorem
- Product measures, Fubini’s theorem
- Elements of functional analysis: Hilbert spaces, orthogonality, bases,
normed spaces, Banach spaces, bounded operators

Notes: We’ll try to discuss problems covering one topic in each class. Familiarity with basic notions and definitions is expected, as well as knowledge of proofs of basic theorems by the end of the course. We’ll use qualifying exam problems – besides textbook problems – from other universities as well.