MAT 533, Real Analysis II

Spring 2021

Christopher Bishop

Professor, Mathematics
Stony Brook University

Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631


MAT 533 is a continuation of MAT 532 and is one of the requried core courses for first year graduate students. We cover an introduction to functional analysis, the Riesz represenation theorem, an introuction to Fourier analysis and distribution theory. If time permits, we will do a few things about probability theory and fractal geometry too.


The syllabus contains information about the textbook and grading.


Real Analysis, 2nd edition, by Gerald Folland

Time and place :

Live online lectures MW 2:40-4:00pm, Zoom lecture link        
Meeting ID: 974 0855 8490, Passcode: textbook author        
For audio only use phone nummber: 646 876 9923 US (New York)        
Find your local number .

Final Exam:

Final exam will be distributed and uploaded via Blackboard. It will be during the scheduled exam period: Thursday, May 13, 11:15-1:45.

The final exam will have 5 problems. The first problem has five subparts that asks for examples of various objects from Chapters 4, 5 and 7. For example, give an example of a meager set with full measure. You do not need to prove your examples are correct. The other four problems require a proof of some sort. You will have a choice of four from five possibilities: two from Chapter 8 (Fourier analysis), two from Chapter 9 (Distributions) and one from Chapter 10 (probability)

You may use the textbook during the exam, but no other sources (books, people, notes,...). Please review the following problems from the text. A few of these (or variations) will appear on the final exam: 8.8, 8.18, 8.32, 8.34, 8.35, 9.10, 9.11, 9.25, 9.33, 10.14, 10.15.

Office hours (via Zoom):

Prof Bishop: Wed 4-5, and by appointment. personal Zoom link .
Meeting ID: 625 069 4063 Passcode: 653434

Jordan Rainone (grader): Monday 1-2pm personal Zoom link .

Lecture notes:

Since we are online, I will try to make lecture slides that that summarize the important material from the text book and occassionally add other remarks:
CHAPTER 4 SLIDES, ANNOTATED The slide plus any corrections, skteches,... I made in class. Much larger file.
CHAPTER 5 SLIDES - Functional analysis
CHAPTER 7 SLIDES - Radon measures
CHAPTER 8 SLIDES - Fourier analysis
CHAPTER 9 SLIDES - Distributions and Sobolev spaces
CHAPTER 10 SLIDES - Topics in Probability Theory
Prof Varolin's notes on existence and uniqueness for ODE
Slides based on Varolin's ODE notes

Machine learning

This subdirectory contains some math paper related to machine learning. The one labeld "SIAMreview" is the most basic and expository. The rest are currently mostly some papers of Charles Fefferman that were recently pointed out to me. Send me link for other papers that you think are interesting and accessible to beginners.

Lecture recordings

Class recordings are available on the class Blackboard page.

Lecture recordings

Previous lectures

Tentative lecture schedule

        Feb 1: Introduction, overview
        Feb 3: 4.3, 4.4 Nets and compactness
        Feb 8: 4.7 Arzela-Asccoli, Tychonoff's theorem, Stone-Weierstrass theorem
        Feb 10: 5.1 Normed vector spaces
        Feb 15: 5.2 Linear functionals
        Feb 17: 5.3 Baire category
        Feb 22: 5.4 Topological vector spaces
        Feb 24: 5.5 Hilbert space
        Mar 1: 5.5 Hilbert space
        Mar 3: 7.1 Positive linear functionals
        Mar 8: 7.2 Regularity and approximation
        Mar 10: 7.3 Dual of C_0
        Mar 15: 8.1 Prelinaries and notation
        Mar 17: 8.2 Convolutions
        Mar 26-27: Midterm (on Chapter 5)
        Mar 24: 8.3 The Fourier transform
        Mar 29: 8.3 The Fourier transform
        Mar 31: 8.4 Summation of Fourier integrals/series
        Apr 5: 8.5 Pointwise convergence
        Apr 7: 8.5 Pointwise convergence
        Apr 12: 8.6 Fourier analysis of measures
        Apr 14: 8.7 Applications to PDE
        Apr 19: 9.1 Distributions
        Apr 21: 9.2 Tempered distributions
        Apr 26: 9.3 Sobolev spaces
        Apr 28: Quick intro on ODE Prof Varolin's notes on ODE
        May 3: 10.1 Basic concepts of probability
        May 5: 10.2 Law of large numbers

Problem sets:

Problem sets should be submitted to the grader. Electronic submissions of LaTexed solutuions is preferable, but scans or photographs of hand-written solutions are acceptable if clearly legible. If you strongly prefer to hand in physical solution sets, contact the grader to make arrangments to drop them at his office, or leave them with the graduate secretary.

My plan to is have problems sets due every other week. My hope was that this was less stressful than weekly assignments, although the total number of problems would be the same. Tentatively, problems sets are due on the Friday following the week(s) when we cover the relevant section. Due dates may change depending on the schedules for other core classes, and the grader.

Late problem sets will be accepted, but with a penalty determined by the grader.

        Due Friday, Feb 12: 4.36 (you may use cited exercises without proof), 4.42, 4.43, 4.68
        Due Friday, Feb 26: 5.5, 5.7, 5.12, 5.18, 5.19
        Due Friday, March 12: 5.27, 5.42, 5.47, 5.53, 5.59, 5.66
        Friday, March 26: No problem set due. The midterm takes its place this week. The exam will be made available on Blackboard at 9am on Friday, March 26, 2021 and should be uploaded in Blackboard (or emailed to the lecturer) by 9pm Saturday, March 27 2021.
        Due Friday, April 9: 7.17, 7.20, 7.22, 7.24 (7.24 needs a correction. See Folland's errata page )
        Due Friday, April 30 (revised from April 23): 8.3, 8.10, 8.11, 8.15, 8.18

Although it is not required, you may wish to consider writing up your solution in TeX, since eventually you will probably use this to write your thesis and papers.

The not too short introduction to LaTex

Additional links

Hugh Woodin, The Continuum Hypothesis, Part I

This gives an introduction to set theory with a discussion of the the role of the axiom of choice and the existence of non-measurable sets.

Hugh Woodin, The Continuum Hypothesis, Part II

This continues the previous article and discusses in what sense the continuum hypothesis can be considered true or false, even through it is formally independent of ZFC.

paper giving careful proof of Banach-Tarski paradox

Wikipedia article on the Banach-Tarski paradox

Wikipedia article on Carleson's a.e. convergence theorem

Wikipedia article on Weierstrass' nowhere differentiable function

Link to Schroder-Bernstein theorem

Link to Freilng's dart argument against CH

Link to history of mathematics

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If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, room 128, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information go to the following website: ]

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Critical Incident Management Statement:

Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Faculty in the HSC Schools and the School of Medicine are required to follow their school-specific procedures.

Student Absences Statement

Students are expected to attend every class, report for examinations and submit major graded coursework as scheduled. If a student is unable to attend lecture(s), report for any exams or complete major graded coursework as scheduled due to extenuating circumstances, the student must contact the instructor as soon as possible. Students may be requested to provide documentation to support their absence and/or may be referred to the Student Support Team for assistance. Students will be provided reasonable accommodations for missed exams, assignments or projects due to significant illness, tragedy or other personal emergencies. In the instance of missed lectures or labs, the student is responsible for insert course specific information here (examples include: review posted slides, review recorded lectures, seek notes from a classmate or identified class note taker, write lab report based on sample data). Please note, all students must follow Stony Brook, local, state and Centers for Disease Control and Prevention (CDC) guidelines to reduce the risk of transmission of COVID. For questions or more information check the unversity website.

Send me email at: bishop at

Link to history of mathematics