An introductory graduate level course including the theory of
integration in abstract and Euclidean spaces, and an introduction
to the basic ideas of functional analysis. Math 5051-5052 form the
basis for the Ph.D. qualifying exam in
analysis. *Prerequisites*: Math 4111, 4171, and 4181, or
permission of the instructor.

Instructor: Ari Stern

Email: stern@wustl.edu

Office: Cupples I 211B

Office Hours: TuTh 11am-12:30pm

TuTh 8:30-9:50am, Mallinckrodt 303. The first class will be on Tuesday, August 29, and the last will be on Thursday, December 7. Class will be canceled for Fall Break (Tuesday, October 10) and Thanksgiving (Thursday, November 23).

Problem sets will be posted
to Canvas
approximately weekly, and students will submit their solutions
electronically by uploading them to Gradescope before the
specified due date and time. You are encouraged to discuss the
homework with your fellow students, but *your final write-up
must be your own*. Please make sure that your solutions are
written clearly and legibly. (Typing up solutions in LaTeX is
encouraged and is valuable practice for mathematical writing later
in your career.)

There will be one in-class midterm exam on Thursday, October 19. The final exam will be held on Friday, December 15, from 1-3pm.

Grades will be based on a weighted average of homework (40%, lowest score dropped), midterm exam (20%), and final exam (40%).

The required textbook for this course is *Real Analysis:
Modern Techniques and Their Applications*, by Gerald
B. Folland (second edition, Wiley, 1999). This book has more than
a few typographical errors, so it's a good idea to check the list
of errata
on Folland's
homepage.

As a supplemental text, I also recommend *Measure,
Integration, & Real Analysis*, by Sheldon Axler, which
is freely available online
(although it is also published by Springer in hardcopy). This is a
very recent text, but it appears to do a nice job of expanding
some of the material that Folland treats more tersely, and giving
concrete motivation where Folland is more abstract.

I have also asked the library to place the following supplemental texts on reserve:

- H. L. Royden,
*Real Analysis*(3rd edition). - E. M. Stein and R. Shakarchi,
*Real Analysis: Measure Theory, Integration, and Hilbert Spaces*. - T. Tao,
*An Introduction to Measure Theory*(based on his freely-available course notes). - R. L. Wheeden and A. Zygmund,
*Measure and Integral: An Introduction to Real Analysis*.

Do not feel obligated to purchase any of these non-required books (although each one is excellent in its own way). I am making them available simply because it can be helpful to see alternative treatments of the same material.

I plan to cover the topics discussed in Folland chapters 1-3, 5, and part of 6:

- Chapter 1: Measure
- Chapter 2: Integration
- Chapter 3: Signed Measures and Differentiation
- Chapter 5: Elements of Functional Analysis
- Chapter 6:
*L*Spaces^{p}

The topics in Chapter 0 are assumed to be prerequisites, as they are typically covered in undergraduate real analysis, and you are encouraged to review them on your own, as needed. (Axler also has a supplement that covers similar prerequisite material.) While Chapter 0 covers the topology of metric spaces, students who have not previously encountered the more general theory of point-set topology are also encouraged to read Chapter 4, which we may reference but will not cover in detail.

All students are expected to adhere to high standards of academic integrity, as specified in the undergraduate student academic integrity policy and graduate school academic and professional integrity policy. Any violations of this policy will be referred to the relevant Academic Integrity Officer. Violations of this policy include, but are not limited to:

- homework collaboration exceeding that permitted in the Homework Assignments section above (e.g., preparing your final write-up side-by-side with another student rather than independently, copying another student's solutions);
- submitting the work of others as your own (e.g., copying solutions found online);
- exam misconduct (e.g., copying another student's exam or knowingly allowing your exam to be copied, giving or receiving unauthorized assistance, use of unauthorized books/notes or other materials, unauthorized use of electronic devices, etc.).

If you have any questions, or are unsure about what is permitted/prohibited by this policy, please ask me.

*In many cases, academic integrity violations are the result
of getting behind in coursework and making bad decisions under
pressure. Keep up with your assignments, ask questions when you
are unsure what is expected of you, and do not give in to the
temptation to cut corners.*

Students experiencing symptoms consistent with COVID-19 or concerned about a possible exposure should contact Habif Health and Wellness Center (314-935-6666) to arrange for testing as indicated. If instructed by Habif to quarantine or isolate, students should notify their instructor as soon as possible by forwarding the email they received from Habif. Any accommodation needs for COVID-related absence not covered in an instructorâ€™s standard course policies should be discussed between the student and instructor.

While on campus, it is imperative that students follow all public health guidelines established to reduce the risk of COVID-19 transmission within our community. The full set of University protocols can be found at https://covid19.wustl.edu/health-safety/.

To encourage in-person attendance and participation, WashU no longer requires that lectures be recorded or otherwise made available outside the classroom, aside from certain exceptional cases: "In the case of excused student absences due to COVID and other factors (e.g., illness, religious holidays, family emergencies, etc.), instructors should develop strategies for providing students access to the fundamental content of a given class session so that students are able to make progress in the course while complying with public health and university guidelines around quarantine and isolation and managing other challenges that disrupt their ability to attend class."

If you cannot attend class due to COVID or other factors similar to those listed above, and if you contact me at least 2 hours prior to the start of class, I will try to arrange for video recording of the lecture. If I am unable to record the lecture (e.g., due to technical difficulties) or do not receive advance notice to do so, I will provide written lecture notes instead.