Spring 2018 MATH 202B 001 LEC

Introduction to Topology and Analysis
Schedule: 
SectionDays/TimesLocationInstructorClass
001 LECTuTh 09:30AM - 10:59AMCory 289Marc A Rieffel26726
UnitsEnrollment Status
4Open
Additional Information: 

Prerequisites: Math 202A and Math 110

Description: Measure and integration. Product measures and Fubini-type theorems. Signed measures; Hahn and Jordan decompositions. Radon-Nikodym theorem. Integration on the line and in Rn. Differentiation of the integral. Hausdorff measures. Fourier transform. Introduction to linear topological spaces, Banach spaces and Hilbert spaces. Banach-Steinhaus theorem; closed graph theorem. Hahn-Banach theorem. Duality; the dual of LP. Measures on locally compact spaces; the dual of C(X). Weak and weak-* topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. Additional topics chosen may include compact operators, spectral theory of compact operators, and applications to integral equations.

Office: 811 Evans

Office Hours: TBA

Recommended Text: Real and Functional Analysis 3rd ed. by Serge Lang, Springer-VerlagBasic Real Analysis by Anthony Knapp. Advanced Real Analysis by Anthony Knapp.  My understanding is that through an agreement between UC and Springer, chapters of the Lang text and the Knapp texts are available for free download by students. See the course webpage for the links to them. For those who have used the text Real Analysis by Folland for Math 202A, that text can also be quite useful for parts of Math 202B. 

 Comments: We will continue on from wherever this Fall's Math 202A ends, to develop the theory of measure and integration.We will also develop more general topology as needed. A major further subject will be an introduction to functional analysis, which consists of methods for dealing with infinite-dimensional topological vector spaces and linear operators on them. This will use both the general topology and the measure and integration that has been covered. It has wide applications, for example to harmonic analysis, partial differential equations, analysis on manifolds, and quantum physics. Some of the items we will discuss are Banach spaces, the closed-graph theorem,  the Hahn-Banach theorem and duality, duals of classical Banach spaces, weak topologies, the Alaoglu theorem, convexity and the Krein-Milman theorem. We will also discuss further related topics if time allows. In my lectures I will try to give well-motivated careful presentations of the material. 

Grading: Grading: I plan to assign roughly-weekly problem sets. Collectively they will count for 50% of the course grade. Students are strongly encouraged to discuss the problem sets and the course content with each other, but each student should write up their own solutions, reflecting their own understanding, to turn in. Even more, if students collaborate in working out solutions, or get specific help from others, they should explicitly acknowledge this help in the written work they turn in. This is general scholarly best practice. There is no penalty for acknowledging such collaboration or help. There will be a final examination on Wednesday May 9, 11:30-2:30 , which will count for 35% of the course grade. There will be a midterm exam, which will count for 15% of the course grade. There will be no early or make-up final examination. Nor will a make-up midterm exam be given; instead, if you tell me ahead of time that you must miss the midterm exam, then the final exam will count for 50% of your course grade. If you miss the midterm exam but do not tell me ahead of time, then you will need to bring me a doctor's note or equivalent in order to try to avoid a score of 0.

Accomodations: Students who need special accomodation for examinations should bring me the appropriate paperwork, and must tell me at least a week in advance of each exam what specific accomodation they need, so that I will have enough time to arrange it.

Course Webpage: math.berkeley.edu/~rieffel

The above information is subject to change.