Math 240A Fall 2021

Math 240A, Fall 2021
Real Analysis

Announcements:

Dec. 1: Next week I will have office hours Mon. 9:00 - 10:30 and 1:00 - 2:00 (in office and zoom), and Qingyuan will have office hours Mon. 2:00 - 3:00 (AP&M 1131 and zoom)
Nov. 30: Here is a practice final
Nov. 22: Office hours this week are on Tuesday by appointment only
Nov. 22: There is no in-person lecture this Wednesday. Instead, see the pre-recorded lecture at https://learn.evt.ai/
Oct. 21: Here is a practice midterm
Oct. 25: This week instead of my Friday office hours I will have office hours on Thursday 1:00-2:00 and 3:00-3:30
Nov. 8: Brandon's office hours this week are W 2-4 and F 9-10. Qingyuan's office hours this week are F 12-2 in AP&M 1131.

Course Information

Lecture: MWF 11:00-11:50 AM in room B402A of the AP&M building (recordings available at https://learn.evt.ai/)
Coronavirus Considerations: Lectures will be recorded and available online, and office hours will be available both in-person and over Zoom. So if you are feeling ill or have been exposed to COVID19, then you are encouraged to not come to class or office hours. Additionally, if I ever feel sick or have been exposed, then classes will occur remotely over Zoom until it is safe for me to return to campus.

Professor: Brandon Seward (pronouns: he/him or they/them)
Email: bseward@ucsd.edu
Office Hours: W 2:00 - 3:30 PM, F 1:00 - 2:30 PM (both in-person in 5739 AP&M and remotely via Zoom ID 967 0420 1807)

TA and Grader: Qingyuan Chen
Email: qic069@ucsd.edu
Office Hours: Th 2:00 - 3:00 PM and 6:00 - 7:00 PM (both in-person in 5829 AP&M and remotely via Zoom ID 971 0835 9812)

Course Description: First course in a three-quarter graduate sequence on real analysis. Topics covered include measures, integration, and differentiation.
Prerequisites: Math 140ABC or equivalent
Textbook: Real Analysis: Modern Techniques and Applications by Gerald B. Folland, 2nd edition. We will cover Chapters 1 - 3.
Textbook Errata: See Folland's homepage

Homework: Homework will be assigned most weeks and due on Fridays by midnight. We will use Gradescope for turning in homework. When registering for gradescope, please register using your "@ucsd.edu" email address and use Entry Code 5V44ZY.

Homework 1 (due Fri. Oct. 1): Ch. 1 problems 1, 2, 3, 4, 5. Also review Sections 0.1, 0.3, 0.5 and 0.6 of the textbook.
Homework 2 (due Fri. Oct. 8): Ch. 1 problems 9, 14, 17, 18(ab), 21 (for the definition of saturated, see Ch. 1 problem 16)
Homework 3 (due Fri. Oct. 15): Ch. 1 problems 25, 26, and Ch. 2 problems 9(b), 10, 11
Homework 4 (due Fri. Oct. 22): Ch. 2 problems 12, 13, 14, 25, 26
Homework 5 (due Fri. Nov. 5): Ch. 2 problems 23, 28, 29, 32, 38 (for 38, assume the functions are complex-valued)
Homework 6 (due Fri. Nov. 12): Ch. 2 problems 51, 55, 59, 60, 63
Homework 7 (due Fri. Nov. 19): Ch. 3 problems 3, 6, 9, 16, 17
Homework 8 (due Fri. Dec. 3): Ch. 3 problems 19, 21, 22, 25(a), 42

Exams: There will be one midterm and one final exam.

Midterm (Solutions): In class Friday Oct. 29. Will cover all of Chapter 1 and the first three sections of Chapter 2. In case of illness or covid exposure you can take the exam remotely (at the same date and time) but you must email me at least an hour in advance to let me know. Practice Midterm.

Final Exam (Solutions): Tuesday Dec. 7 from 11:30 AM to 2:30 PM in our normal lecture room. Will cover everything learned in the class. In case of illness or covid exposure you can take the exam remotely (at the same date and time) but you must email me at least an hour in advance to let me know. Practice Exam

Grading: Your course grade will be computed from the maximum of the following two weighted formulas:

30% homework + 30% midterm + 40% final
40% homework + 60% final

Course Schedule (items in gray are tentative)

Week	Monday	Wednesday	Friday
0			Sept. 24 Introduction σ-Algebras
1	Sept. 27 σ-Algebras	Sept. 29 σ-Algebras Measures	Oct. 1 (HW 1 Due) Measures Outer measures
2	Oct. 4 Outer measures Borel measures on the real line	Oct. 6 Borel measures on the real line	Oct. 8 (HW 2 Due) Borel measures on the real line
3	Oct. 11 Measurable functions	Oct. 13 Measurable functions Integration of nonnegative functions	Oct. 15 (HW 3 Due) Integration of nonnegative functions Integration of complex functions
4	Oct. 18 Integration of complex functions	Oct. 20 Integration of complex functions	Oct. 22 (HW 4 Due) Modes of convergence
5	Oct. 25 Modes of convergence Product measures	Oct. 27 Product measures	Oct. 29 Midterm Exam (Solutions) Practice Exam
6	Nov. 1 The n-dimensional Lebesgue integral	Nov. 3 The n-dimensional Lebesgue integral	Nov. 5 (HW 5 Due) The n-dimensional Lebesgue integral
7	Nov. 8 The n-dimensional Lebesgue integral Integration in polar coordinates	Nov. 10 Integration in polar coordinates Signed measures	Nov. 12 (HW 6 Due) Signed measures The Lebesgue--Radon--Nikodym theorem
8	Nov. 15 The Lebesgue--Radon--Nikodym theorem	Nov. 17 The Lebesgue--Radon--Nikodym theorem Complex measures	Nov. 19 (HW 7 Due) Complex measures Differentiation on euclidean space
9	Nov. 22 Differentiation on euclidean space	Nov. 24 no in-person lecture Recording, Notes Differentiation on euclidean space	Nov. 26 Thanksgiving Holiday
10	Nov. 29 Differentiation on euclidean space Functions of bounded variation	Dec. 1 Functions of bounded variation	Dec. 3 (HW 8 Due) Functions of bounded variation
11	Tuesday Dec. 7, 11:30AM-2:30PM in normal lecture room Final Exam (Solutions) Practice Exam