Zheng-Chao Han
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.
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
640:501 or equivalent.