# Math 5052

## Measure Theory and Functional Analysis II, Spring 2013

### Basic Information

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

### Lectures

Lectures will be held MWF 10-11am in Rudolph Hall 102. The first class will be on Monday, January 14, and the last will be on Friday, April 26. Class will be canceled for Martin Luther King, Jr. Day (Monday, January 21) and Spring Break (week of March 10-16).

### 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.)

Benjamin Passer (bpasser@math.wustl.edu) is responsible for homework grades and solutions.

• HW1 [pdf]. Due Wednesday, January 23.
• HW2: Folland, Chapter 8, Exercises 8, 11a, 13, 14, 15. Due Wednesday, January 30.
• HW3: Folland, Chapter 8, Exercises 16, 18, 19, 20. Due Wednesday, February 6.
Errata: In Exercise 16a, replace f by fk. In Exercise 18c (top of p. 256), replace right by left.
• HW4: Folland, Chapter 8, Exercises 26, 27, 28, 31, 32. Due Wednesday, February 13.
• HW5: Folland, Chapter 8, Exercises 34, 36, 37, 43, 45. Due Wednesday, February 20.
Erratum: In Exercise 43, replace e-|x|/2 by ½e-|x|.
• HW6: Folland, Chapter 9, Exercises 2, 8, 9, 13, 15. Due Wednesday, February 27.
• HW7: Folland, Chapter 9, Exercises 30, 32, 33, 36, 37. Due Wednesday, March 27.
Hint for Exercise 32: Split the integral defining ||f||(s) into the regions |ξ| ≤ R and |ξ| > R, and show that the inequality holds for sufficiently large R.
• HW8: Folland, Chapter 10, Exercises 1, 2, 3, 5, 6. Due Wednesday, April 3.
• HW9: Folland, Chapter 10, Exercises 11, 14, 16, and Chapter 7, Exercise 2. Due Wednesday, April 10.
• HW10: Folland, Chapter 7, Exercises 1, 4, 5, 6. Due Wednesday, April 17.
Note: For Exercise 6, skip the case where X is a one-point compactification.
• HW11: Folland, Chapter 11, Exercises 1, 2, 3, 4. Due Wednesday, April 24.

### Exams

There was one in-class midterm exam, held on Friday, March 8.

The real analysis qualifying exam (which takes the place of a final exam) is scheduled for Wednesday, May 8, from 10am-1pm, in Cupples I, 199. The qualifying exam will be cumulative, covering material from both semesters of Math 5051-5052.

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

### Course Outline

I plan to cover the following topics from Folland:

• Chapter 8: Elements of Fourier Analysis
• Chapter 9: Elements of Distribution Theory
• Chapter 10: Topics in Probability Theory
• Chapters 7.1 and 11.1: Radon Measures and Haar Measure