Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631
M-W 9:45-11:05, Physics P-124
Real Analysis, 2nd Edition by Gerald Folland, Wiley
We will cover the first five chapters of Folland's text book. The first three chapters deal with measure theory on abstract metric and topological spaces, although the construction of Lebesgue measure on the real line is the principal example. Chapter 4 deals with some point set topology. Because of overlaps with the topology courses, I will only review parts of Chapter 4, although we will cover the Arzela-Ascoli theorem and the Stone-Weierstrass theorem in detail. Chapter 5 is in introduction to Banach and Hilbert spaces and we will prove the basic results such as the Hahn-Banach theorem, open mapping theorem and uniform boundedness principle. If time permits, we will start on Chapter 6, which discusses L^p spaces.
Students are responsible for the material in Chapters 1-5. I will probably not have time to cover every detail of every section in class, but I will try to discuss the main ideas, examples and proofs in lecture and will try to address any other questions that are raised. From time to time I will also try to present some examples and applications not discussed in the textbook (but these will not be covered on the problem sets or exams).
If you are interested in set theory and logic, one online textbook that covers much of the same material as Folland's book, but with a greater emphasis on set theory (e.g., Borel and non-Borel sets, analytic sets, transfinite induction,...) is available online: Real Analysis by Bruckner, Bruckner and Thomson
I expect the students in MAT 532 will have had a solid undergraduate analysis course and be familiar with some point set topology (open, closed, compact sets), Riemann integration, metric spaces, infinite sets, ... Here at Stony Brook, the two undergraduate classes MAT 320 and MAT 324 are the usual prerequisites; the latter class constructs Lebesgue measure on the real line and has some overlap with the first few weeks of this course. Chapter 0 of our textbook briefly reviews the material that will be assumed later in the text.
Here are the midterms and finals for a 2-semester undergraduate course from Rudin's 'Principles of Mathematical Analysis'. These should give you an idea of what would be good to know entering this course: midterm 1 , final 1 , midterm 2 , final 2 ,
Problem sets, Midterm and Final will each count for one third of the course grade. Final is Dec 12, 2002 11:15am-1:45pm in P-124, our usual room.
Mon Aug 22: First class, Chapter 0, 1.1 Introduction
Wallace-Bolyai-Gerwien theorem
Banach-Tarski paradox
An interview with Gerald Folland (our textbook author)
Wed Aug 24: 1.2 sigma-algebras, 1.3 Measures
A preprint of mine with a brief introduction to descriptive set thory and some explicit examples
of non-Borel sets.
Mon Aug 29: 1.4 Outer measures
Wed Aug 31: 1.5 Borel measures on the real line
Mon Sept 5: 2.1 No class -- Labor day
Wed Sept 7: Finish 1.5
Mon Sept 12: 2.1 Measurable functions
Wed Sept 14: 2.2 Integration of non-negative functions
Mon Sept 19: 2.3 Integration of complex functions
Wed Sept 21: 2.4 Modes of convergence
Mon Sept 26: 2.5 Product measures,
lecture slides for today's online class.
video recording mjoc#$M8
Wed Sept 28: 2.6 The n-dimensional Lebegue integral,
An Elementary Proof of a Theorem of Johnson and Lindenstrauss
Mon Oct 3: 3.1 Signed measures (class starts 9am, today only, due to Math Day lectures)
Wed Oct 5: 3.2 The Radon-Nikodym theorem
Mon Oct 10: No class -- Fall break
Wed Oct 12: Midterm (Chapters 1 and 2)
Mon Oct 17: 3.3 Complex measures, Start 3.4
Wed Oct 19:
3.4 Differentiation on Euclidean space
Mon Oct 24:
3.5 Functions of bounded variation
Wed Oct 26:
4.1 Topological spaces, 4.2 Continuous maps
Mon Oct 31:
4.3 Nets, 4.4 Compact sets
Wed Nov 2: 4.6 Two compactness theorems (Tychonoff
and Arezela-Ascoli)
Mon Nov 7: 4.7 The Stone-Weierstrass theorem
Wed Nov 9: 5.1 Normed vector spaces
Mon Nov 14: 5.2 Linear functionals
Wed Nov 16: 5.3 The Baire category theorem
Mon Nov 21 5.4 Topological vector spaces
Wed Nov 23: No class -Thanksgiving break
Mon Nov 28: 5.5 Hilbert spaces
Short proof that c_0 (sequences tending to zero) is not complemented in
l^\infty (bounded sequences).
Survey article on complmented subspaces. Mentions famous 1971
result of Lindenstrauss and
Tzafriri that a Banach space is a Hilbert space iff
every closed subspace is complemented.
Wed Nov 30: Continue 5.5
Mon Dec 5: Last class, make-up, review
Wed Dec 12: Final exam, 11:15am-1:45pm, Romm P-124 Physics (usual room)
Problem sets are due at begining of class Wednesdays. Paper copies may be handed in at class, or emailed to the grader before class begins.
Due Aug 31: 1.2: 3, 5 and 1.3: 8, 10, 12, 14My office hours will be after class, M-W 11:15-12:45 in my office, 4-112 in the Math Building, and by appointment. Meeting in person or on Zoom can be arranged. Questions by email are also welcome.
Ocassionally, I may be away on a class day. In these circumstances, I will probably hold class live via Zoom, or post a pre-recorded lecture covering the scheduled material. I will let you know ahead of time on a base-by-case basis.
The grader is Runjie Hu.
Although it is not required, you may wish to consider writing up your problem solutions in TeX, since eventually you will probably use this to write your thesis and papers.
The not too short introduction to LaTex
Hugh Woodin, The Continuum
Hypothesis, Part I
Hugh Woodin, The Continuum
Hypothesis, Part II
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 Schroeder-Bernstein theorem
Link to Freilng's dart argument against CH
Link to mathematical biographies
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