Real Analysis II

Homework

This can be found on the Gitea server

Syllabus

Course description

This course is really a continuation of the course on real analysis at the 400 level. In this course we will discuss various special functions of a real variable, including Fourier Series representations of periodic functions. We will also cover some of the theory of standard calculus in several variables, but with a more advanced viewpoint than previously encountered. Finally, we will learn about other approaches to some of the ideas in calculus, which are ultimately better suited to abstraction: the Riemann–Stieltjes integral and the Lebesgue integral. The Lebesgue integral is of particular importance because, among other things, it naturally makes the functions with a finite integral into a complete space. In fact, one can even view it as the completion of the space of step functions with respect to this norm.

We will use the text Principles of Mathematical Analysis by Walter Rudin.

Course objectives

By the end of this course, students should be able to:

• Prove results corresponding the Lebesgue and Riemann–Stieltjes integrals.
• Compute the Fourier Series of a periodic $L^2$ function and understand its meaning.
• Understand calculus of several variables from the advanced perspective of differential forms.
• Utilize the completeness of $L^p$ spaces to prove basic results about measurable functions.

Course Outline And Topics

More than likely, we will not cover topics in this order. Moreover, some material will be emphasized less than others. Decisions will be somewhat dependent on the interests of the students.

1. Chapter 6. The Riemann-Stieltjes Integral
2. Chapter 7. Sequences and Series of Functions
3. Chapter 8. Some Special Functions
4. Chapter 9. Functions of Several Variables
5. Chapter 10. Integration of Differential Forms
6. Chapter 11. The Lebesgue Theory

Grading procedure

• Two take home exams (30% each)
• Homework (25% total)
• Gitea (15% total)

Take-home exams are expected to be completed individually without the aid of your classmates; you are welcome to use the book and your notes, but you may not consult resources online. Homework will be assigned weekly. You are highly encouraged to work on the homework problems together with your classmates as this promotes learning, however, you should each write your final solutions independently. The Gitea item corresponds to an online server. You will upload solutions to some of your homework problems to the site (you must create an account first). You should think of this as a sort of ``collaborative book'' for the course. I will contribute, and you will as well. Together, we will create a nice summary of the topics, theorems, problems and their proofs throughout the semester. For this, it will be expected that you use LaTeX to type the math, but the whole document is a variant of Markdown. We will cover the usage on the first day.

Important dates

Makeups

If an emergency arises which requires you to miss an exam, I must be made aware at least two hours prior to the start time of your exam. Note: proper documentation will be required before a makeup arrangement is considered.

Academic dishonesty/misconduct

Academic misconduct includes, but is not limited to, cheating, plagiarism and forgery, and soliciting, aiding, abetting, concealing, or attempting such acts. Plagiarism may consist of copying, paraphrasing, or otherwise using written or oral work of another without proper acknowledgment of the source or presenting oral or written material prepared by another as ones own. At minimum, cheating will result in that assignment receiving a grade of zero.

Official communication

Your SIUE student e-mail account is the official method to communicate between you and your instructor. Official communication will not be sent to your personal e-mail (yahoo, wildblue, gmail etc.).

Accessible Campus Community & Equitable Student Support

Students needing accommodations because of medical diagnosis or major life impairment will need to register with Accessible Campus Community & Equitable Student Support (ACCESS) and complete an intake process before accommodations will be given. Students who believe they have a diagnosis but do not have documentation should contact ACCESS for assistance and/or appropriate referral. The ACCESS office is located in the Student Success Center, Room 1270. You can also reach the office by e-mail at myaccess@siue.edu or by calling (618)~650-3726. For more information on policies, procedures, or necessary forms, please visit the ACCESS website at www.siue.edu/access.

Disclaimer

This syllabus is subject to change by the instructor if deemed necessary for the benefit of student learning or to correct errors and omissions.