Math 110.416: Honors Analysis 2, Spring 2020

Course description:
This course continues 110.415 Honors Analysis I, with an emphasis on the fundamental notions of modern analysis. Topics covered will include functions of bounded variation, Riemann-Stieltjes integration, Riesz representation theorem, along with measures, measurable functions, and the lebesgue integral, properties of Lp spaces, and Fourier series.

Where and when:
Lecture will be held Mondays and Wednesdays 1:30-2:45 in Bloomberg 176 (NOTE WE CHANGED THE ROOM). Discussions will be held Fridays 1:30-2:20 in Gilman 277.

Instructor: Alex Mramor, amramor1 'at'
Office hours 3-5 Thursdays at Krieger hall 311
Teaching Assistant: Junyan Zhang, jzhan182 'at'

Main text:
Real Analysis, by Carothers, Cambridge, ISBN:9780521497565. Please let me know if you are having trouble getting the book.

There will be an in class final worth 40% of your grade, an in class midterm worth 20% ( Update, 2/19: we will have the test 3/25 ), weekly 20-30 minute long quizzes in discussion worth 25%, and weekly homework of 7-10 problems assigned Wednesdays and due the following Fridays in discussion worth the last 15%.

The new (post COVID) grading scheme: 40 percent homework, 60 percent take home final.

Academic integrity:
The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

Report any violations you witness to the instructor. You may consult the associate dean of students and/or the chairman of the Ethics Board beforehand. Read the “Statement on Ethics” at the Ethics Board website for more information.

Disability accommodations policy:
Students with documented disabilities or other special needs who require accommodation must register with Student Disability Services. After that, remind the instructor of the specific needs at two weeks prior to each exam; the instructor must be provided with the official letter stating all the needs from Student Disability Services.

Course schedule and assignments (updated often! - below my numbering is off at some point because I skipped spring break):

Week 1:
We'll begin the new semester with the last material covered in honors analysis 1 in the fall semester: Chapter 11, "the space of continuous of functions," out of Carothers.

Homework due Friday: Nothing due, but spend the week reviewing what you covered in math 415, and pick out and try a few problems from the book.

Week 2:
We covered chapter 13 on Monday, and Wednesday will be from chapter 14.

Homework due this Friday (posted Monday, 2/3/20 - typically homework will be assigned Wednesday and due the Friday after. l'll list homework under the week it is due for future reference): problems 47, 52, 54, 55, 56 from Chapter 11 of Carothers. NEW HOMEWORK IS POSTED BELOW.

Week 3:
We covered (nearly) all of what we wanted to cover in chapter 14, up through "integrators of bounded variation"

Homework due Friday (posted Wednesday, 2/5, due Friday 2/14: Valentines day): do problems 1, 4, 6, 12 from chapter 13, and problems 1, 2, 5, 8, 11, 22 from chapter 14.

Week 4:
We wrapped up chapter 14 (the fundamental theorem of calculus parts 1 and 2) and did chapter 15, up through Dirichlet and Fejer kernels.

Homework due Friday (posted Wednesday, 2/12, due Friday 2/21): do problems 10, 25, 26, 27, 36, 38, 40, 41, 46 out of chapter 14.

Week 5:
In chapter 16 we covered the section "Lebesgue outer measure," skipped (we'll come back to this) "Riemann integrability," and started the section "measurable sets." We gave the definition (Carother's definition, not the original one) of measurable set, and showed the collection of measurable sets was an algebra of sets.

Homework due Friday (posted Wednesday, 2/19, due Friday 2/28): do problems 4,5, 7, 8, 11 out of chapter 15 (yes, a light assignment this week!)

Week 6:
We essentially wrapped up all we wanted to cover in chapter 16, except for Lebesgue's theorem (concerning the Riemann integral). I also skipped Vitali covering theorem, which we may come back to later.

Highlights from this week include the following: we showed that the (Lebesgue) measurable sets were a so-called sigma algebra and that the outer measure was countably additive on the collection of measurable sets, giving that, restricting to the set of measurable sets, the outer measure fulfilled our wishlist for defining area in the buildup to Lebesgue integration. We also showed there were nonmeasurable sets however, much to Our chagrin. Then we discussed the structure of Lebesgue measurable sets - namely that they are up to null sets G delta/ F sigma sets, and showed our/Carothers definition of measurability agreed with Lebesgue/Cartheodory's definition.

Homework due Friday (posted Wednesday, 2/26, due Friday 3/6): do problems 3,4,5,6,8,9,16,17,20, 21 out of chapter 16.

Week 7:
We wrapped up chapter 16 with Lebesgues theorem, we were going to then start discussing measurable functions, but that plan was dashed by COVID-19.

Homework due Friday (posted Wednesday, 3/4, due Friday 3/13 - the thirteenth!!! ): do problems 39, 41, 42, 44, 48, 49, 73, 74, 75, 77.

Week 8:
Our first week back from break and moving entirely to remote. We covered measurable functions/ chapter 17 and have started chapter 18, up to monotone convergence theorem.

Homework that was previously due on 3/13 is due 3/25.

Week 9:
We wrapped up chapter 18.

Homework due this Friday (posted 3/25, due 4/3): do problems 1, 3, 4, 5, 6, 12, 13, 41, 42, 44, 45, 46, 50. Yi pointed out to me there was a problem with question 5; suppose alpha is in (0,1) in that problem.

Week 10:
We did what we wanted out of chapter 19 this week: convergence in measure and L^p spaces.

Homework due this Friday (posted 4/2, due 4/12 (yes, a Saturday, since I was a little late)): do problems 2, 3, 4, 5, 6, 7, 15, 24, 27, 31 from chapter 18 (you aren't done with problems from chapter 18, though).

Week 11:
We nearly got through chapter 20 this week, stopping short of absolute continuity.

Homework due this Sunday (posted 4/10, due 4/19 (I was late again even moreso so let's make the due date Sunday and have only 9 problems)): do problems 34, 36, 37, 38, 39, 43, 47, 53, 55 out of chapter 18.

Week 12:
We got through chapter 20 and covered what we wanted out of Carothers. For the remainder I'm thinking to direct us towards Sobolev spaces, but if there are any requests I'm happy to hear them.

Homework due this Thursday (posted 4/21, due 4/30): do problems 1, 2, 5, 10, 12, 16, 18, 21, 22, 30 out of chapter 19.

Week 13:
We took a jaunt into chapter 5 of Evan's PDE book, introducing Sobolev spaces and the Sobolev inequalities. An indication on why these spaces were useful in PDE was also given.