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Some important announcements

Math 7210: Real Analysis II

2021 Fall, Auburn University

Contacts

Lecture Instructor Dr. Le Chen lzc0090@auburn.edu
Class Time and Room TR, 9:30 AM -- 10:45 AM PARKR 326
Office hours TR, 11:00 AM -- 11:50 AM, PARKR 203, or via appointment/Zoom upon request

Course description

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions.[1] Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

This course is the second course of the two-semester sequential courses -- Math 7200. In Math 7200, we have covered Chapters 0-3 of the textbook.

Textbook

Coverage

This course will cover topics such as signed measures, Lebesgue-Radon-Nikodym theorem, function of bounded variation, topological vector spaces, Hilbert spaces, \(L^p(\mathbb{R}^d)\) spaces. Depending on the progress, we will follow mostly most parts of the following about five chapters of the text book starting from Section 2.4:

Prerequisite

Students obligations and tips

This is a demanding course and it requires a great deal of work from your side. In order to successfully master the material and complete the course, you are expected to

Note: The syllabus was created in Oct. 2021, and it is subject to changes during the semester.

Homework

Test and exam

Attendance

Assessment

Slides

Chapter/Section Slides Slides
Chapter 2: Integration presentation compact
2.1. Measurable functions pres. comp.
2.2. Integration of nonnegative functions pres. comp.
2.3. Integration of complex functions pres. comp.
2.4. Modes of convergence pres. comp.
2.5. Product measures pres. comp.
2.6. The n-dimensional Lebesgue integral pres. comp.
2.7. Integration in polar coordinates pres. comp.
Chapter 3: Signed measures and differentiation presentation compact
3.1. Signed measures pres. comp.
3.2. The Lebesgue-Radon-Nikodym theorem pres. comp.
3.3. Complex measures pres. comp.
3.4. Differentiation on Euclidean space pres. comp.
3.5. Functions of bounded variation pres. comp.
Chapter 4: Point set topology presentation compact
4.1. Topological spaces pres. comp.
4.2. Continuous maps pres. comp.
4.3. Nets pres. comp.
4.4. Compact spaces pres. comp.
4.5. Locally compact Hausdorff spaces pres. comp.
4.6. Two compactness theorems pres. comp.
4.7. The Stone-Weierstrass Theorem pres. comp.
4.8. Embedding in Cubes pres. comp.
Chapter 5: Elements of functional analysis presentation compact
5.1. Normed vector spaces pres. comp.
5.2. Linear functionals pres. comp.
5.3. The Baire category theorem and its consequences pres. comp.
5.4. Topological vector spaces pres. comp.
5.5. Hilbert spaces pres. comp.
Chapter 6: Lp spaces presentation compact
6.1. Basic theory of Lp spaces pres. comp.
6.2. The dual of Lp pres. comp.
6.3. Some useful inequalities pres. comp.
6.4. Distribution functions and weak Lp pres. comp.
6.5. Interpolation of Lp spaces pres. comp.

Tentative schedule

Gradescope

Face Covering Policy

We will follow the university policy regarding face covering: 

Students enrolled in this course are required to wear a face covering that covers the nose and mouth
while inside the classroom, laboratory, faculty member offices, or group instructional spaces.
Failure to comply with this requirement represents a potential violation of Code of Student Conduct
and may be reported as a non-academic violation.

Please consult the Auburn University Classroom Behavior Policy at

for additional details.

Honor code

Accessibility

Your success in this class is important to me. We will all need accommodations because we all learn differently. If there are aspects of this course that prevent you from learning or exclude you, please let me know as soon as possible. Together we will develop strategies to meet both your needs and the requirements of the course.

I encourage you to visit the Office of Accessibility to determine how you could improve your learning as well. You can register and make a request for services from the Office of Accessibility. In this case, please do inform me of such requests. See the following link for more information:

Harassment and Discrimination


Auburn University is committed to providing a working and academic environment free from prohibited
discrimination and harassment and to fostering a nurturing and vibrant community founded upon the
fundamental dignity and worth of all its members. Auburn University prohibits harassment of its
students and employees based on protected classes and works to eliminate prohibited behavior from
its academics and employment through corrective measures and education.

The Office of AA/EEO oversees compliance with the Policy Prohibiting Harassment of Students, the
Policy Prohibiting Harassment of Employees, and the Policy on Sexual and Gender-Based Harassment and
Other Forms of Interpersonal Violence.

Protected classes are race, color, sex (which includes sexual orientation, gender identity, and
gender expression), religion, national origin, age, disability, protected veteran status, or genetic
information.

Auburn University also prohibits retaliation against any individual for opposing a practice he/she
reasonably believed to be discriminatory; for filing an internal or external complaint, grievance,
or charge; or for participating in any investigation or proceeding, in accordance with Auburn
University's policies.

Feedback

Acknowledgement

© Le Chen, Auburn, 2021.