Spring 2026
Zheng-Chao Han
Course Description:
This course is a continuation of 640:501 from Fall 2025. The goal is to give an introduction to core topics in real and functional analysis that every professional mathematician should know. The choice of topics will be somewhat influenced by what is covered in the 501 course in fall 25, but will focus on concepts and techniques that have broad applications to different areas of mathematics. More concretely, we will discuss different modes of convergences and their applications (including weak convergence); manifestations of completeness from different perspectives (Lp spaces and also some elementary Banach space theory and basic theorems involving bounded linear operators); compactness and applications; and some elementary aspects of spectral theory of (compact) linear operators. A substantial part of analysis is to estimate the magnitudes of quantities via various inequalities, which will constitute a part of the syllabus. We may also cover some aspects of analysis on manifolds, time permitting/interest from students. All the general ideas will be illustrated in some concrete contexts of applications. Although many of the topics to be covered are not on the syllabus of the written qualifying exam, they provide ample space for students to witness the applications of the ideas and tools learned in 501 in a variety of contexts, and to practice problem solving skills.
Textbook:
We will continue to use the 501 text, Real Analysis: Modern Techniques and Their Applications, by J. Folland (Wiley-Interscience; 2nd edition, 1999. ISBN-10: 0471317160)
as a general reference and source for problem sets, but will often draw material from other sources. Below is a short list of likely additional sources. Rutgers students can get free e-books of the Springer textbooks via the Rutgers library link.
Richard F. Bass, Real analysis for graduate students, 2nd edition, 2013, ISBN-13: 978-
1481869140. https://www.math.wustl.edu/ victor/classes/ma5051/rags100514.pdf.
• Haïm Brezis, Functional analysis, Sobolev spaces and PDEs, Springer, New York,
2011. Will only touch on early chapters.
• H. L. Royden and P. M. Fitzpatrick, Real Analysis, Fourth Edition (ISBN 978-0-13-
143747-0), Pearson Education, 2010.
• Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction
to Real Analysis, Marcel Dekker, 1977, or its 2nd edition, by CRC Press in 2015.
• William Ziemer, Modern Real Analysis, Springer Cham, 2017, https://doi.org/10.1007/978-
3-319-64629-9. Also https://www.math.purdue.edu/ torresm/pubs/Modern-real-analysis.pdf
Prerequisites:
640:501 or equivalent.
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Spring 2025
Michael Kiessling
Course Description:
The course 502 is a continuation of Fall’s 501, but also offers an outlook on applications in mathematical physics. We will pick up where the course 501 ended, so 501 is a pre-requisite. The material is mostly from Folland’s book. We begin with some selected material from section 3 (Signed measures and differentiation), then hop to section 5 (Elements of functional analysis) and continue with sections 6 (L p spaces), 7 (Radon measures), 8 (Elements of Fourier analysis) and 9 (Elements of distribution theory). Since the material of 502 is no longer tested on the written qualifying exam, we have some liberty to substitute some of the above by some selected material of sections 10 (Probability theory) or 11 (Haar measure, Hausdorf measure), depending on the preferences of the class. I plan to occasionally supplement Folland’s book by “hand outs†of some functional analysis material that I have typed-up.
Textbook:
"Real Analysis'' by Folland, 2nd ed.
Prerequisites:
640 501 or equivalent
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Spring 2024
Dennis Kriventsov
Course Description:
This is a continuation of Real Analysis 1. We will cover further topics in measure theory and integration, the Riesz representation theorem, the Fourier transform, distributions, and differentiation.
Textbook:
TBA
Prerequisites:
501
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Spring 2023
Yanyan Li
Course Description:
The course 502 is a continuation of Fall’s 501. We will pick up where the course 501 ended, so 501 is a pre-requisite.
Textbook:
“Real Analysis” (2nd ed) by G.B. Folland
Prerequisites:
501
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Spring 2022
Fioralba Cakoni
Course Description:
It is the second part of real analysis.
Textbook:
Real Analysis by Folland
Prerequisites:
501
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Spring 2021
Michael Kiessling
Course Description:
The course 502 is a continuation of Fall’s 501, but also offers outlook on applications in other fields of mathematics, and in mathematical physics and engineering. We will pick up where the course 501 ended, so 501 is a pre-requisite. The material is mostly from Folland’s book. We begin with some selected material from section 3 (Signed measures and differentiation), then hop to section 5 (Elements of functional analysis) and continue with sections 6 (L p spaces), 7 (Radon measures), 8 (Elements of Fourier analysis) and 9 (Elements of distribution theory). Since the material of 502 is no longer tested on the written qualifying exam, we may also look into some selected material of sections 10 (Probability theory) and 11 (Haar measure, Hausdorf measure) at the end of the course. I plan to occasionally supplement Folland’s book by “hand outs” of typed-up material. Given the pandemic situation, the course is currently planned to be taught in synchronous remote mode; a room in Hill Center has been booked for in-class instruction in case the situation improves significantly.
l
Textbook:
“Real Analysis” (2nd ed) by G.B. Folland
Prerequisites:
Mathematics 501 or equivalent
Schedule of Sections:
16:640:502 Schedule of Sections
Previous Semesters:
Spring 2020 Prof. Sussmann
Spring 2016 Prof. Sussmann
Spring 2015 Prof. Nussbaum
Spring 2014 Prof. Chanillo
Spring 2013 Prof. Carlen
Spring 2012 Prof. Chanillo
Spring 2011 Prof. Nussbaum
Spring 2010 Prof. Han
Spring 2009 Prof. Carlen
Spring 2007 Prof. Goodman
Spring 2004 Prof. Goodman