# Lecture Notes

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math104-s22:start

# Math 104: Introduction 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)
Lecture: TuTh 9:30A-10:59A at Evans 3

Final : Wed, May 11, 11:30A - 2:30P • Evans 3

• Online Lecture Zoom
• Online Discussion Discord
• Virtual Group Discussion SpatialChat
• Gradescope: homework submission, entry code GEDJP3

GSI: Jacobo, Evans 747. zoom office.

• M 2-5pm; T: 12-2; W: 12-3pm (remote); F: 2-4pm
• First two weeks all above slots are for remote access.

## Syllabus

What do we cover in this class? It will consist of three parts

• Sequence and Limits
• Metric space Topology
• Differentiation and Riemann Integral

How to learn this course? One should

• 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.
• participate 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.
• participate in discord discussion. You can ask question about lecture and homeworks, and post your homework solutions here for others to comment.
• [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, and whatever you think is relevant for this course.

How do we grade? Our grade will consist of

 participation (20%) + midterms (30%) + final (50%).

Weekly homework is part of participation, and is not graded based on correctness.

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.
• make correction to the homework, and upload to your student homepage.

We will have two in-class midterms, and the one with the lower grade will be dropped.

The final is accumulative, 3-hour in class.

## Textbook

We will be using three textbooks

Some other lecture notes might be useful

## Journal

We will first cover sequence and limits, mainly following Ross and Tao-I

Then, we will cover metric space topology and continuous functions, where we will follow Rudin and Tao-II.

Finally, we will cover integration and differentiation, following Rudin.

 Lecture Date Day Content Reading Notes and Videos Lecture 1 Jan 18 Tue Number system $\R$ Ross §1 - §4 note, video Lecture 2 Jan 20 Thu Completeness Axioms and Sequence Ross §4, §7 note, video, HW1 Lecture 3 Jan 25 Tue Basic properties and examples of limits Ross §9 note, video Lecture 4 Jan 27 Thu monotone seq, limsup, liminf Ross §10 note, video,HW2 Lecture 5 Feb 1 Tue Cauchy Seq, Subseq Ross §10, §11 video Lecture 6 Feb 3 Thu Subseq, limsup liminf Ross §11, §12 video,HW 3 Lecture 7 Feb 8 Tue more on limsup/liminf , Series Ross 12, 14 video Lecture 8 Feb 10 Thu More on series. Rearrangements Rudin, Ch 3 video , HW4 Lecture 9 Feb 15 Tue review Lecture 10 Feb 17 Thu midterm 1,stat (no Office Hour today) 2021-spring No HW Lecture 11 Feb 22 Tue Exam review. Begin metric space. Lecture 12 Feb 24 Thu closure, limit points Ross 13 video HW 5 Lecture 13 Mar 1 Tue More metric spaces. Continuous function Pugh Ch 2 video Lecture 14 Mar 3 Thu Compactness video HW6 Lecture 15 Mar 8 Tue Theorems about Compactness. Connectedness. Rudin 2.41, 2.45-2.47 video Lecture 16 Mar 10 Thu Connectedness. Rudin 4.13 - 4.27 video , HW 7 Lecture 17 Mar 15 Tue Discontinuity. Uniform Continuity Rudin Ch 4 video Lecture 18 Mar 17 Thu Sequence and Convergence of functions Rudin Ch 7.1, 7.2 video HW 8 Spring Recess Mar 22 Tue Spring Recess Mar 24 Thu Lecture 19 Mar 29 Tue Differentiation. Mean Value Theorem video(partially recorded) Lecture 20 Mar 31 Thu L'hopital rule. video No HW Lecture 21 Apr 5 Tue Midterm 2 solution Lecture 22 Apr 7 Thu Review of Midterm 2. Taylor Theorem video , HW 9 Lecture 23 Apr 12 Tue Power Series. Riemann integral Ross 32 video Lecture 24 Apr 14 Thu Riemann integral, weight function Ross 33, 35 no video, HW 10 Lecture 25 Apr 19 Tue More on integrability Rudin Ch 6 video, note Lecture 26 Apr 21 Thu Integration and Differentiation Rudin Ch 6 video note, HW 11 Lecture 27 Apr 26 Tue uniform convergence and $\int, d/dx$ Rudin Ch 7, p151-153 video, note Lecture 28 Apr 28 Thu review of 2021sp final video RRR May 3 Tue RRR May 5 Thu
math104-s22/start.txt · Last modified: 2022/05/02 14:24 by pzhou