### Math 132: Real Analysis II

Professor Francis Su

Fall 2022

Course Webpage: https://math.hmc.edu/su/math132/

My Office: Shanahan 3416

Drop-In Hours:
Fridays 1:30-2:30pm (on Zoom)
Mondays 4-5pm (on Zoom, or in-person with notice).
The Zoom sessions require a passcode, which is on the course Sakai page.
Also available by appointment.

My Email: (my last name) at math.hmc.edu

Grader: Maxwell Thum (mthum at g.hmc.edu)

This course is a continuation of the ideas in Analysis (Math 131). There’s a lot of interesting and deep ideas that you’ll enjoy learning about. Topics include: Riemann-Stieltjes integration, function spaces, equicontinuity, uniform convergence, the inverse and implicit function theorems, differential forms, and an introduction to Lebesgue integration and measure theory.

## Required Text

Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill, 3rd edition. We will cover Chapter 6 onward. There are also many other books on analysis that you may wish to consult in the library, around the QA300 area.

## Coursework

Homeworks will be assigned and collected weekly. There will be one midterm and one final exam. Each component (homework, midterm, final) is worth at least 30% of your final grade, with the “best” component worth 40%.

It is helpful to remember that course grades are just intended to assess what you have learned. But they are a not a reflection of your potential ability in mathematics, nor your worth as a human being. I believe everyone in the class is fully capable of mastering this material. Questions are valued, even simple ones because they can lead to profound ideas. Exploration is encouraged, especially risk-taking in trying out things that may not work, because this is how we learn.

The learning you are doing in this class takes place in a larger framework of school and life. Even if I am excited about teaching and you are excited about learning, work is not the most important thing, and sometimes life outside the classroom can take precedence. I can be somewhat flexible in accommodating requests for homework extensions and absences for other important events. Please make these requests 24 hours in advance, if possible.

Similarly, ‘success’ by whatever measure is not the most important thing in this course either. Every assessment of your work in this class is a measure of progress, not a measure of promise. Joy, wonder, productive struggle, having your mind expanded—these are more important!

## Honor Code

The HMC Honor Code applies in all matters of conduct concerning this course. Though cooperation on homework assignments is encouraged, you are expected to write up all your solutions individually. Thus copying is prohibited. You should understand your solutions well enough to write them up yourself. It is appropriate to acknowledge the assistance of others; if you work with others on a homework question, please write their names in the margin. Part of the fun of this course is the struggle, as well as the joy of discovering a solution for yourself. Please note: using solutions found online or solutions of prior students will be regarded as a violation of the HMC Honor Code and will be handled accordingly. I encourage you instead to talk to me or the tutors or each other!

## Lecture Notes and Zoom

Lecture Notes are available here. If you cannot come to class, but would like to join via Zoom, go to the Sakai page for the course. Keep your camera off and you won’t appear on the Zoom recording. (Also, on Zoom, I will not be able to hear or see the chat.)

## Reviewing Analysis I

These lectures were taped in 2010 and you may find them valuable for review:

Real Analysis Lectures, Spring 2010.

## Homeworks

Due on Gradescope on Tuesdays at 10pm.

Some of you may find LaTeX helpful in typesetting your homework. If so, there is a LaTeX class for homework here.

All HW’s refer Rudin’s Principles of Mathematical Analysis.

Fill out this information sheet if you haven’t already.

Here is Homework 1 and its LaTeX code.

And if you haven’t read this article yet, do it!

On each assignment I may assign “reading problems” which are problems you should read and reflect on in Rudin, but you do not have to do them. They are marked with an ‘R’ on the assignment.

Here is HW #2 and the TeX code.

Remember to follow the guidelines for good mathematical writing.

Do Chapter 6 (6, R7, 8, 10abc, 11, 15 ).

Hint on 10a: for a concave up function, its values always lie
below the secant line between two endpoints.

Hint on 15: yes, that last inequality is strict.

Read Chapter 7, theorems 7.1-7.15.

Do Chapter 7 ( 1, 2, 3, R4, 5, 6 ).

In Problem 5, you might notice how this compares with the statement of the M-test.

Read the rest of Chapter 7.

Do Chapter 7 ( 7, 8, 9, R11, R14, 15, 16 )

Hints: On 7.8, thm 7.12 can still be useful.

On 7.9, use eps/2 argument. Also, where it asks “Is the converse true?” it is asking this:

If {f_n} is a sequence of continuous functions on a set E, and if
lim f_n(x_n) = some f(x)
for every sequence of points x_n in E such that x_n->x and x in E, must f_n converge uniformly to f?

If you believe the answer is NO, then all you have to do is give an specific example of f_n and f where the convergence is not uniform.  I’d think of as simple an example as possible.

The midterm for the class will be made available Sat Oct 8 and due the following week, Wed Oct 12 (with some flexibility as needed.)

The midterm for the class has been posted on Gradescope, due before Fall Break.

Do the problems on this handout. Here’s the TeX file.

On Problem 9.2, you can (and should) assume that A is invertible.

Do Chapter 9 ( 3, 5, 6, 8, 9 ).

It may be helpful on 9.9 to remember that a connected set cannot be written as the disjoint union of two non-empty open sets.

Do Chapter 9 ( 11, 13, 16, 17ab, 17cd ).

For these problems, you can assume knowledge of calculus for taking derivatives of things like sin(x), etc.

Do Chapter 9 ( 18 (worth double), 19 (worth double), 20 ).

In 9.18(b), you can interpret the analogous question as “Find where the Jacobian is non-zero, and interpret in light of the inverse function theorem. Show that f is not globally 1-1.”

Optional Group Assignment (in groups of 1-5 people). Only 1 person needs to upload, but they should remember to tag the other group members.

Redo Questions 2 and 3 on the exam, but concentrate on writing up the simplest, most elegant solutions. Appeal to theorems we proved in class or in the book. Discuss with others how to simplify the arguments.

If you missed x points on the exam, re-doing these questions will earn you:

• x/3 points back for any solid attempt.
• x/2 points back for especially clear, elegant, and correct solutions.

There’s no HW over Thanksgiving Break. Next (and last) homework due Tue Dec 6.

Do Chapter 10 ( 15, 20, R21 ) and

Problem A. Prove Theorem 10.20a in your own words.

Problem B. Prove Theorem 10.20b in your own words.

Problem C. Find a differential 2-form $\omega$ in $R^4$ such that
$\omega \wedge \omega$ is not zero.

(The upshot of Problem C is to have you realize that while the wedge product of a basic form with itself is zero, the same is not necessarily true for other forms.)

Below this line, all homeworks are TENTATIVE. This means they are likely to be assigned, but there is no guarantee that they will until you see them moved above.