Math 240A, Fall 2021

Real Analysis

## Announcements:

##

## Course Information

Real Analysis

- 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.

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.

## Course Schedule (items in gray are tentative)

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

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

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 |