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.

My 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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