Math 140C, Spring 2021
Foundations of Real Analysis III


Announcements:


Course Information

Professor: Brandon Seward (pronouns: he/him/his)
Email: bseward@ucsd.edu
Lecture: MWF 11:00-11:50 AM via Zoom (navigate to https://ucsd.zoom.us/j/99316078691 or use Meeting ID 993 1607 8691)
Lecture Notes
Office Hours via Zoom: W 12:00 - 1:00 PM & 5:00 - 7:00 PM via Zoom (navigate to https://ucsd.zoom.us/j/99656671941 or use Meeting ID 996 5667 1941)
Office Hour Notes
Note: Office hours and lecture have different Meeting IDs!

Teaching Assistant: Srivatsa Srinivas (pronouns: he/him/his)
Email: scsriniv@ucsd.edu
Discussion: Mon. 7:00-7:50 PM via Zoom (navigate to https://ucsd.zoom.us/j/91672235630 or use Meeting ID 916 7223 5630)
Office Hours via Zoom (morning): Tu 8:00 - 9:00 AM via Zoom (navigate to https://ucsd.zoom.us/j/99983476457 or use Meeting ID 999 8347 6457)
Office Hours via Zoom (evening): Th 6:00 - 7:00 PM via Zoom (navigate to https://ucsd.zoom.us/j/91009741292 or use Meeting ID 910 0974 1292)

Course Description:
Third course in a rigorous three-quarter sequence on real analysis. Topics include: differentiation of functions of several real variables, the implicit and inverse function theorems, the Lebesgue integral, infinite-dimensional normed spaces.

Textbook: Principles of Mathematical Analysis by Walter Rudin, 3rd edition. We will cover most of Chapters 9 and 11.

Lecture:
Lecture will occur live on Zoom (use meeting ID: 993 1607 8691 or navigate to https://ucsd.zoom.us/j/99316078691). Videos of our lectures will be posted on our Canvas page under the Media Gallery tab. Additionally, the Lecture Notes I write in class will be updated shortly after each lecture.

Community Interaction: Typically meeting in the classroom allows students the opportunity to get to know one another and discuss the course material together. Since class meetings will not occur, as a substitute we have a discussion forum on Piazza that everyone is encouraged to use. Our class piazza page is here: http://piazza.com/ucsd/spring2021/math140c.

Homework: Homework will be assigned regularly and due on Fridays at 9:00 PM. No late homework will be accepted. On each assignment, a few problems will be graded for correctness, while the others will be graded simply for completion. We will use Gradescope for turning in homework. When registering for gradescope, please register using your "@ucsd.edu" email address and use Entry Code 86D28X.

Homework 1 (due Fri. April 9):  Chapter 9 problems 2, 3, 5, 8, 10 (just prove the first claim), 13, 15
Homework 2 (due Fri. April 16): Chapter 9 problems 6, 7, 9, 11, 12 (but don't show density), 14, 27 (Hint for 14c: approximate Ɣ by its derivative)
Homework 3 (due Fri. April 30): Chapter 9 problems  16, 17(abc), 19, 21, 23, 28, 30(ab) (Hint for 21b: write f as (x+y) times a quadratic polynomial)
Homework 4 (due Fri. May 7)
Homework 5 (due Fri. May 14): Chapter 11 problems 1, 2, 3, 4, 5
Homework 6 (due Fri. May 28):
Chapter 11 problems 6, 7, 8, 9, 10, 12 (Hint for #7)
Homework 7 (due Fri. June 4): Chapter 11 problems 11, 13, 14, 16, 17, 18

Exams: The midterm exams will be taken at home and solutions will be uploaded to Gradescope. The exams will be open-book and open-note, but the use of online resources and help from other humans is forbidden. If cheating is suspected, students will be required to have a one-on-one zoom meeting to solve similar problems. Students found to be cheating will receive a 0 on their exam.
Grading: Your final grade will be the maximum of the following two weighted averages:
Special Accommodations: Students requiring accommodations should provide an OSD letter of certification and OSD accommodation recommendation to me as soon as possible.


Course Schedule (items in gray may change)

Week
Monday
Wednesday
Friday
1
March 29
Linear transformations
March 31
Linear transformations
April 2
Linear transformations
2
April 5
Differentiation
April 7
Differentiation
April 9 (HW 1 Due)
Differentiation
The contraction principle
3
April 12
The inverse function theorem
April 14
The inverse function theorem
The implicit funciton theorem
April 16 (HW 2 Due)
The implicit function theorem
Determinants

4
April 19
Determinants
April 21
First Midterm

Practice A, Practice B
(No lecture)
April 23
Determinants
Derivatives of higher order
Differentiation of integrals

5
April 26
Differentiation of integrals
Set functions

April 28
Set functions
Construction of the Lebesgue measure

April 30 (HW 3 Due)
Construction of the Lebesgue measure
6
May 3
Construction of the Lebesgue measure
May 5
Construction of the Lebesgue measure
Measure spaces
Measurable functions

May 7 (HW 4 Due)
Measurable functions
Simple functions

7
May 10
Integration
May 12
Integration
May 14 (HW 5 Due)
Integration
8
May 17
Integration

May 19
Second Midterm
Practice A, Practice B
(no lecture)
May 21
Comparison with the Riemann integral
9
May 24
Comparison with the Riemann integral
Integration of complex functions
Functions of class L^2

May 26
Functions of class L^2
May 28 (HW 6 Due)
Functions of class L^2
10 May 31
Memorial Day Holiday
June 2
Functions of class L^2
(Not on final) Existence of non-measurable sets
June 4 (HW 7 Due)
Review
11
Friday June 11, 11:30AM-2:30PM
Final Exam
Practice A, Practice B