Math 745 - Analysis II

Course Outline - Spring 2002

Text: J. MacDonald and N. Weiss, Real Analysis, Academic (ISBN 0-12-742830-5)

Suppl. Reading: T. Apostol, Mathematical Analysis; I. Natanson, Theory of Functions of a Real Variable; H. Royden, Real Analysis; W. Rudin, Real and Complex Analysis.

Topics Sections

Week 1 Set theory. Axiom of choice 1.1 - 1.2

Week 2 Cardinality and algebras of sets (notes) 1.3 - 1.4

Week 3 Topology: metric spaces and completeness 7.1, 7.2, 7.4

Week 4 Separation properties. Tietze's theorem. Connectivity 7.6 - 7.8

Week 5 Review of advanced calculus. Riemann and Riemann -

Stieltjes integration 2.1 - 2.6

Week 6 Lebesgue measure and integration 3.1 - 3.8

Week 7 Abstract measure and integration. Midterm 4.1 - 4.5

Week 8 Abstract measure and integration continued 4.6 - 4.9

Week 9 Banach and Hilbert spaces. Projection theorem 9.1 - 9.3

Week 10 Lebesgue and Sobolev spaces. Applications (notes) 9.4, 9.5

Week 11 Review of Fourier series and transforms 11.1 - 11.3

Week 12 Harmonic analysis. Applications (notes) 11.5

Week 13 Applications: dynamical systems and ergodic theory 12.1 - 12.2

Week 14 Introduction to wavelets. Review 11.6, 11.7

Final Examination

(prepared by D. Blackmore - 1/4/02)