Math 240B, Winter 2022
Real Analysis II


Announcements:


Course Information

Lecture: Pre-recorded lectures are available at https://learn.evt.ai/. Also see the Lecture Notes.
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: MWF 11:00 - 11:50 AM at Zoom ID 967 0420 1807 (https://ucsd.zoom.us/j/96704201807)

TA and Grader: Qingyuan Chen
Email: qic069@ucsd.edu
Office Hours: Thurs. 5:00 - 7:00 pm at Zoom ID 964 2787 7552 (https://ucsd.zoom.us/j/96427877552)

Course Description: Second course in a three-quarter graduate sequence on real analysis. Topics covered include point set topology, functional analysis, L^p spaces, and Radon measures.
Prerequisites: Math 240A and Math 140ABC or equivalent
Textbook: Real Analysis: Modern Techniques and Applications by Gerald B. Folland, 2nd edition. We will cover most of Chapters 4 through 7.
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 6P4K5Z.

Homework 1 (due Fri. Jan. 14):  Ch. 5 problems 2, 3, 6, 12(abcd), 17, 18 and Ch. 6 problem 5
Homework 2 (due Fri. Jan. 21):  Ch. 5 problem 7 and Ch. 6 problems 2, 7, 8, 11, 13, 15 (in #13, "separable" means there exists a countable dense subset)
Homework 3 (due Fri. Jan. 28): 
Homework 4 (due Fri. Feb. 11): 
Homework 5 (due Fri. Feb. 18): 
Homework 6 (due Fri. Feb. 25):
Homework 7 (due Fri. March 4):
Homework 8 (due Fri. March 11):

Exams: There will be one midterm and one final exam.
Grading: Your course grade will be computed from the following weighted formula: 30% homework + 30% midterm + 40% final


Course Schedule (items in gray are tentative)

Week
Monday
Wednesday
Friday
1
Jan. 3
5.1 Normed vector spaces
Jan. 5
5.1 Normed vector spaces
5.2 Linear functionals
Jan. 7
5.2 Linear functionals

2
Jan. 10
6.1 Basic theory of L^p spaces
Jan. 12
6.1 Basic theory of L^p spaces
Jan.  14 (HW 1 Due)
6.2 The dual of L^p
3
Jan. 17
Martin Luther King Jr. Holiday

Jan. 19
6.2 The dual of L^p
6.3 Some useful inequalities
Jan. 21 (HW 2 Due)
6.3 Some useful inequalities
6.4 Distribution functions and weak L^p
4
Jan. 24
6.4 Distribution functions and weak L^p
5.3 The Baire Category Theorem and its consequences
Jan. 26
5.3 The Baire Category Theorem and its consequences
Jan.  28 (HW 3 Due)
5.5 Hilbert spaces
5
Jan. 31
5.5 Hilbert spaces
Feb. 2
4.1 Topological spaces
Feb. 4
Midterm Exam
(no lecture)
6
Feb. 7
4.1 Topological spaces
4.2 Continuous maps
Feb. 9
4.2 Continuous maps
Feb.  11 (HW 4 Due)
4.3 Nets
4.4 Compact spaces

7
Feb. 14
4.4 Compact spaces
4.5 Locally compact Hausdorff spaces

Feb. 16
4.5 Locally compact Hausdorff spaces
Feb. 18 (HW 5 Due)
4.5 Locally compact Hausdorff spaces
4.6 Two compactness theorems
8
Feb. 21
President's Holiday


Feb. 23
4.6 Two compactness theorems
5.4 Topological vector spaces
Feb.  25 (HW 6 Due)
5.4 Topological vector spaces
9
Feb. 28
7.1 Positive linear functionals on C_c(X)
March 2
7.1 Positive linear functionals on C_c(X)
March 4 (HW 7 Due)
7.1 Positive linear functionals on C_c(X)
7.2 Regularity and approximation theorems
10 March 7
7.2 Regularity and approximation theorems
March 9
7.3 The dual of C_0(X)
March  11 (HW 8 Due)
7.3 The dual of C_0(X)
11
Monday March 14, 11:30AM-2:30PM
Final Exam