## MATH 131C: Topics in Analysis

• Course description: This is the third quarter of a sequence on Analysis, completing the honors sequence as well as the regular Math 131AB.
We will review the Riemann integral in Rn and then discuss Lebesgue measure, the Lebesgue integral, several convergence theorems, the relation of Lebesgue integration with Riemann integration, and finally differentiation theory. We will cover chapters 1 and 2 of the Stein-Shakarchi textbook in detail, as well as selected topics from chapters 3 and 4. Proofs will be important, and the best attitude to bring to this class is not to believe any theorem that you don't know how to prove.

Instructor: Burt Totaro.
E-mail: totaro@math.ucla.edu.
Office Hours: 3-3:50 M and 2-2:50 F, in my office MS 6136.

TA: Maria Ntekoume (mntekoume@math.ucla.edu).
TA office hour: 4:30-5:30 M and 3-4 F, in MS 6139.

• Lecture: MWF 11-11:50, MS 5118. Discussion: Tu 11-11:50, MS 5147.

• Textbook: E. M. Stein and R. Shakarchi, Real Analysis, Princeton University Press (2005), ISBN 0-691-11386-6.

• Prerequisite: Math 131B or 131BH is required.

Midterm Exams: We will have two midterm exams. The dates are Monday, April 23 and Monday, May 21. There will be no makeup exams.
Final Exam: The final exam is on Wednesday, June 13, 2018 from 11:30 AM to 2:30 PM. You must take the final to pass the class! If you have a documented reason that you are unable to take the final, you will receive an Incomplete.

Evaluation:
• Every exam will include at least one problem taken from the homework, possibly with minor variations.
• It is your responsibility to know how to do the problems. Practicing that is an essential part of studying for the exams.
• A grade of 'F' will be assigned to any student who misses the final. Incompletes are reserved for those who have completed all of the work for the class, including both midterms, but who, for a legitimate, documented reason, miss the final.
• Exams (or copies) will be returned, but I will keep copies (or originals) of the exams, as required by the math department.
• Grading: Grades will be assigned based on the higher of the following two schemes:
10% homework + 25% first midterm + 25% second midterm + 40% final
10% homework + 35% (best of two midterms) + 55% final
• 4/23 - First midterm exam.
• 5/21 - Second midterm exam.
• 5/28 - Memorial Day holiday. No class.
• 6/13 - Final exam. The final will be from 11:30 AM to 2:30 AM on Wednesday, June 13.
• Course web page: http://www.math.ucla.edu/~totaro/131c.1.18s/index.html

• Homework 1 (due April 10, 2018): page 37, exercises 2, 4, 5, 6, 7.

• Homework 2 (due April 17, 2018): page 37, exercises 8, 9, 11, 12. Page 46, problem 1.

• Homework 3 (due April 31, 2018): page 37, exercises 13, 16, 18, 22, 29. Page 46, problem 5.

• Homework 4 (due May 8, 2018): page 37, exercise 28. Page 89, exercises 6, 8, 10, 11, 15.

• Homework 5 (due May 15, 2018): page 89, exercises 23, 24. Page 95, problem 2.

• Homework 6 (due May 29, 2018): page 144, exercises 2, 3, 4. Page 193, exercises 1, 2, 4.

• Homework 7 (due June 5, 2018).

Tentative schedule of lectures, in terms of sections in the book:
4/2: Intro. 4/4: 1.1. 4/6: 1.2.
4/9: 1.2. 4/11: 1.3. 4/13: 1.3
4/16: 1.3. 4/18: 1.4. 4/20: 1.4.
4/23: Midterm 1. 4/25: 1.4. 4/27: 2.1.
4/30: 2.1. 5/2: 2.1. 5/4: 2.2.
5/7: 2.2. 5/9: 2.3. 5/11: 2.3.
5/14: 2.3. 5/16: 2.4. 5/18: 3.1.
5/21: Midterm 2. 5/23: 3.1. 5/25: 3.1.
5/28: Memorial Day holiday. 5/30: 4.1. 6/1: 4.1.
6/4: 4.2. 6/6: 4.2. 6/8: Review.