Math 5051

Measure Theory and Functional Analysis I, Fall 2021

Basic Information

Instructor: Ari Stern
Email: stern@wustl.edu
Office: Cupples I 211B
Office Hours: TuTh 10:30am-12pm

Lectures

Lectures will be held TuTh 8:30-9:50am in Cupples II L009. The first class will be on Tuesday, August 31, and the last will be on Thursday, December 9. Class will be canceled for Fall Break (Tuesday, October 12) and Thanksgiving (Thursday, November 25).

Per university policy, lectures will be recorded and made available for later viewing. Recordings may be accessed through Canvas, using the Kaltura Media Gallery link in the navigation bar. This policy is intended "to both support student learning and to facilitate absences necessitated by good public health practice." Please feel free to use these recordings to review material from lectures you have attended, or in case you must stay home due to illness or not passing self-screening, but do not use them as a substitute for regular in-person attendance.

Homework Assignments

Problem sets will be posted to Canvas approximately weekly, and students will submit their solutions electronically by uploading them before the specified due date and time. You are encouraged to discuss the homework with your fellow students, but your final write-up must be your own. Please make sure that your solutions are written clearly and legibly. (Typing up solutions in LaTeX is encouraged and is valuable practice for mathematical writing later in your career.)

Devin Akman (akman@wustl.edu) is responsible for grading the homework assignments.

Exams

There will be one in-class midterm exam on Thursday, October 21. The final exam will be held on Friday, December 17, from 1-3pm.

Grading

Grades will be based on a weighted average of homework (40%, lowest score dropped), midterm exam (20%), and final exam (40%).

Required and Supplemental Texts

The required textbook for this course is Real Analysis: Modern Techniques and Their Applications, by Gerald B. Folland (second edition, Wiley, 1999). This book has more than a few typographical errors, so it's a good idea to check the list of errata on Folland's homepage.

As a supplemental text, I also recommend Measure, Integration, & Real Analysis, by Sheldon Axler, which is freely available online (although it is also published by Springer in hardcopy). This is a very recent text, but it appears to do a nice job of expanding some of the material that Folland treats more tersely, and giving concrete motivation where Folland is more abstract.

I have also asked the library to place the following supplemental texts on reserve:

• H. L. Royden, Real Analysis (3rd edition).
• E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces.
• T. Tao, An Introduction to Measure Theory (based on his freely-available course notes).
• R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis.

Do not feel obligated to purchase any of these non-required books (although each one is excellent in its own way). I am making them available simply because it can be helpful to see alternative treatments of the same material.

Course Outline

I plan to cover the topics discussed in Folland chapters 1-3, 5, and part of 6:

• Chapter 1: Measure
• Chapter 2: Integration
• Chapter 3: Signed Measures and Differentiation
• Chapter 5: Elements of Functional Analysis
• Chapter 6: L^p Spaces

The topics in Chapter 0 are assumed to be prerequisites, as they are typically covered in undergraduate real analysis, and you are encouraged to review them on your own, as needed. (Axler also has a supplement that covers similar prerequisite material.) While Chapter 0 covers the topology of metric spaces, students who have not previously encountered the more general theory of point-set topology are also encouraged to read Chapter 4, which we may reference but will not cover in detail.

Academic Integrity

All students are expected to adhere to high standards of academic integrity, as specified in the undergraduate student academic integrity policy and graduate school academic and professional integrity policy. Any violations of this policy will be referred to the relevant Academic Integrity Officer. Violations of this policy include, but are not limited to:

• homework collaboration exceeding that permitted in the Homework Assignments section above (e.g., preparing your final write-up side-by-side with another student rather than independently, copying another student's solutions);
• submitting the work of others as your own (e.g., copying solutions found online);
• exam misconduct (e.g., copying another student's exam or knowingly allowing your exam to be copied, giving or receiving unauthorized assistance, use of unauthorized books/notes or other materials, unauthorized use of electronic devices, etc.).

If you have any questions, or are unsure about what is permitted/prohibited by this policy, please ask me.

In many cases, academic integrity violations are the result of getting behind in coursework and making bad decisions under pressure. Keep up with your assignments, ask questions when you are unsure what is expected of you, and do not give in to the temptation to cut corners.

Catalog Description

An introductory graduate level course including the theory of integration in abstract and Euclidean spaces, and an introduction to the basic ideas of functional analysis. Math 5051-5052 form the basis for the Ph.D. qualifying exam in analysis. Prerequisites: Math 4111, 4171, and 4181, or permission of the instructor.