Math 105: A second course to real analysis (2022 Spring)

Instructor: Peng Zhou
Email: pzhou.math@berkeley.edu
Office: Evans 931, zoom office
Office Hour: TuTh, 11:10 - 12:30, Friday 4-4:50 (zoom, by appointment. send me a message on discord to let me know)

Final : May 12: 3-6pm, Evans 3

• Online Lecture: Zoom
  
• Online Q & A: Discord
  
• Notes and homework sharing: Students Area
• Gradescope Entry Code: WYE5BN

What this course is about? It is about 3 topics:

• Lebesgue measure / integral.
• Multi-dimensional calculus. Differential Forms. Stokes Theorem.
• Fourier Analysis.

How do we run this course? We will have

• read the textbook and then come to lecture. In lecture, I will focus on motivations and examples, and key steps in the proofs, but I will leave many details as exercises for you to fill in.
• in class group discussion. Since we don't have a discussion session, this is the chance for you to know your classmates. We will have 2-3 person small groups, and have discussion 20-30 minutes. For the first two weeks, we will use spatialchat.
• Discord discussion: a place to discuss lecture contents, and homework questions.
• [NEW] Student homepage. Our course has an area where you can create your own homepage. In your homepage, you can share your class notes, homework solutions, whatever you want to share and think will be helpful to you or others.

How is this course graded? The grade will consist of participation in discussions (50%) and final (exam or essay and presentation, TBD) (50%).

It is hard to quantify participation, and I don't want to assign grade based on how many sentence you say. If one has to set a criterion, let's say, each week, you should

• ask and answer at least one question on discord.
• post your homework solution to to discord
• in discord, comment at least two other student's homework posting.
• You will also submit the homework to gradescope, where the grader or myself will give you comments.
• You will be also given an in class quizzes, those will not be collected and graded, but will be used for self-diagnosis.

To measure performance, we will have a final essay and a 5-10 minutes presentation. You can choose a topic of your interests, read and write up an introduction survey, and record a video presentation. It is a bit vague for now, we can discuss how to make it work. (If this is too hard to figure out, we will just have a usual final exam)

Here is what the letter grade should mean

• C: understand the definition and examples, understand the statement of the theorems, and able to tell whether a statement is true or false.
• B: achieve C and, able to able to solve problem of level of textbook exercises.
• A: achieve B and, understand the logic structure of the theorems, able to outline the key steps of the proofs. Able to explain clearly, in homework, in discussions. Actively participate in discussion (asking or answering questions).

So, say again, where should I submit homework?
The homework will be posted at the end of each week on this website, the homework will contain questions that you have encountered in class discussion, and some more exercises from Tao.

You will submit the homework three times:

1. First, to the discord server, the #homework channel, so that other people can comment on your homework, give suggestions.
2. Second, submit to the gradescope, so that our grader will give you more detailed comments.
3. Finally, after you receive the graded homework back on Gradescope, you can make corrections, and post the corrected HW on your homepage. This is to keep a record for you and your classmates.

Although the homework is graded based on completion, I do hope you give each problem a serious thought. If you see a problem being too easy, you can just sketch the main steps; if you see a problem too hard, you can describe your failed attempt (or even better, ask for suggestions first on discord).

• Charles Pugh, Real Mathematical Analysis, second editiion. springer link
• Tao, Analysis II. (third edition) springer link
• (Optional) Tao, An introduction to measure theory. link to pdf on author's website.

You should be able to download the first textbooks using the above links, once you login to your UC Berkeley account. The third one is advanced undergraduate level and first year graduate level book.

We will first cover Lebesgue measure and Lebesgue integral. This is covered Chapter 6 of Pugh, Chapter 7 and 8 in Tao. We will spend about about half the course doing this. Then, we consider multivariable calculus, with the goal of understanding the Stokes theorem, this is Chapter 5 of Pugh and Chapter 6 in Tao. Finally, we consider Fourier series, Chapter 5 in Tao.

You can find the homeworks in the folder Homeworks, or click on the links below.

For the sake of grading, we will still have an in class final. It will be an open book exam. Exam will be on May 12 (Thursday, 3-6pm, Evans 3). It will consist of 5 problems, where you will decide if a statement is true or false; if true give a proof, if false, give a counter-example.

In addition to the final, if you are interested in writing an essay (post on your homepage), or give a 15-min presentation (in the class day of class), here are some topics and you are free to choose your own.

• The Banach-Tarski paradox (see wikipage)
• Some counter-examples in analysis (see the book in discord)
• What is Fast Fourier transformation? What is wavelet transformation? How to relate to uncertainty principle? (use wiki)
• How to put a measure on the space of Brownian motion?