# Math 5051

## Measure Theory and Functional Analysis I, Fall 2012

### Basic Information

Instructor: Ari Stern
Email: astern@math.wustl.edu
Office: Cupples I, 211B
Office Hours: MWTh 11am-12pm

### Lectures

Lectures will be held MWF 10-11am in Cupples I, 199. The first class will be on Wednesday, August 29, and the last will be on Friday, December 7. Class will be canceled for Labor Day (Monday, September 3), Fall Break (Friday, October 19), and Thanksgiving Break (Wednesday, November 21 through Friday, November 23).

### Homework Assignments

Problem sets will be posted here weekly, on Wednesdays, and will be collected the following Wednesday at the beginning of class. You are encouraged to discuss the homework with your fellow students, and to collaborate on problems, 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.)

Will Ward (wbward@math.wustl.edu) is responsible for homework grades and solutions.

• HW1 [pdf]. Due Wednesday, September 5.
• HW2: Folland, Chapter 1, Exercises 7, 8, 13, 14, 17, 18, 19. Due Wednesday, September 12.
• HW3: Folland, Chapter 1, Exercises 22a, 26, 27, 29, 30, 31, 33. Due Wednesday, September 19.
• HW4: Folland, Chapter 2, Exercises 3, 4, 8, 9, 10, 12, 13. Due Wednesday, September 26.
• HW5: Folland, Chapter 2, Exercises 19, 20, 21, 26, 34, 36. (Hint for Exercise 34: Use the fact that a sequence of real numbers converges iff every subsequence has a further subsequence converging to the same limit.) Due Wednesday, October 3.
• HW6: Folland, Chapter 2, Exercises 39, 42, 44, 46, 50, 56, 59. Due Wednesday, October 10.
• HW7: Folland, Chapter 3, Exercises 2, 4, 5, 9, 13, 16, 17. Due Wednesday, October 17.
• HW8: Folland, Chapter 5, Exercises 3, 5, 6, 7, 12ab, 13, 17, 19. Due Wednesday, November 7.
• HW9: Folland, Chapter 5, Exercises 31, 32, 37, 44, 46, 47, 55, 56. Due Wednesday, November 14.
• HW10: Folland, Chapter 5, Exercises 55, 56, 57, 58, 59, 63, 67. Due Wednesday, November 28.
• HW11: Folland, Chapter 6, Exercises 7, 9, 11, 12, 18, 19, 21. Due Wednesday, December 5 Friday, December 7.

### Exams

There was one in-class midterm exam, held on Friday, October 26.

The final exam was held on Monday, December 17, from 10:30am-12:30pm.

Grades will be based on a weighted average of homework (40%, lowest two scores 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.

In addition, I have 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: Lp Spaces

The topics in chapters 0 and 4 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.

### 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 418, or permission of the instructor.