# Math 5051 - Measure Theory and Functional Analysis I

## Fall 2008

room meeting times
Cupples I, 111 TuTh 2:30p - 4:00p

instructor phone # office e-mail office hours
Renato Feres 5-6752 Cupples I 17 feres@math.wustl.edu Mo 12:00 - 2:00 / We 12:00 - 2:00

Course Plan: the following are topics we hope to cover in the Fall:

• Measurable spaces and the construction of measures;
• The Hahn-Jordan decomposition;
• Lebesgue integration and Lebesgue-Stieltjes measure;
• Product measures, n-dimensional Lebesgue integral, Fubini's theorem;
• Modes of convergence and convergence theorems;
• Signed and complex measures, the Lebesgue-Radon-Nikodym theorem;
• Bounded variation and differentiation of measures;
• Topological spaces, compact and locally compact spaces, compactness theorems;
• Stone-Weierstrass theorem;
• Normed vector spaces, linear functionals and the Hahn-Banach theorem;
• Baire category theorem, open mapping theorem, closed graph theorem, uniform boundedness principle;
• Hilbert spaces;
• Lp spaces;
Topics for the Spring semester: elements of Fourier analysis, the theory of distributions, Sobolev spaces and elliptic partial differential equations, spectral theory of operators, topics in probability theory.

Text: Gerald B. Folland. Real Analysis - Modern Techniques adn Their Applications, second edition, Wiley-Interscience, 1999.

This is, unfortunately, a somewhat expensive text. The changes to the second edition are relatively minor, so a copy of the older edition will serve just fine.

Grades: will be based on weekly homework assignments, a mid-term exam, and a final exam, according to the scale 60%, 20%, 20%, respectively. Midterm will be (due) on October 16 and Final will be (due) on December 16. Grade scale: A [85%,100%], B [70%, 85%), C [55%, 69%), D [45%, 55%), F [0%, 45%).

Please have your homeworks turned in by the deadline, which is 4:00pm of the due date. Hand it in directly to me, and not to the grader, Wei Deng.

Homework Assignments