MATH 746 Theory of Functions of a Real Variable I Fall 2023

Rob Thompson Hunter College
Wednesday 7:30-9:20pm Room: Hunter East 921
August 25-December 20, 2023

e-mail: robert.thompson@hunter.cuny.edu
Office: 902 HE Hours: M 2:30-4:00, W 1:30-3:30, and by appointment

Recorded lectures

• Minkowski's Inequality
• Holder's Inequality

The final exam will be an in-class exam held on Wednesday, December 20, 7:30-9:30pm

The final exam will be comprehensive, meaning that all the topics we covered are elegible to appear on the exam. Here is a brief summary of the topics you are responsible for:

• Outer measure, Legbesque measure on the real line, counting measure
• Measurable functions and the Lebesque integral
• Convergence theorems for Lebesgue integration
• Banach spaces, specifally L^p spaces, including L^1, L^2,Holder's theorem, Minkowksi's theorem

Here is a small smattering of exercises from the book which you can work on. We can discuss these later on, before the final.

• 2B, pg. 38/18,27
• 2D, pg. 60/5
• 3A, pg. 84/5,9,11
• 3B, pg. 99/6,8
• 7A, pg. 199/1,9
• 7B, pg. 208/8,9

Here is Exam Two. It is a take home exam, due on Tuesday, December 12. You are expected to do you own work.

Here are solutions to Exam two

.

We had Exam One on Wednesday, 10/25. The exam covered Chapters Two and Sections 3A.

Here are the Exam One solutions (complete)

We had a quiz Wednesay, Septemeber 20. Here are the solutions.

• The quiz.
• Here are solutions to the quiz.

Desciption of the Course

This is a rigorous introduction to Real Analysis, at the graduate level. We will study Lebesgue measure on the real line, measurable functions, and Lebesgue integration. We will define L_p spaces and study their basic properties. Time permitting, we may demoonstrate some applications to PDEs and Fourier Analysis.

Texts:

• required: Measure, Integration, and Real Analysis by Sheldon Axler, published by Springer in their Open Access Program. The book is free and available online at this link. You may also purchase a hard copy from Springer for $59.99, which is a good price for a book of this publishing quality.

Here are some additional references

Real Analysis, by H. L. Royden

The Way of Analysis, Robert Strichartz

Real and Complex Analysis, Walter Rudin

Prerequisites:

MATH 351 or any undergraduate real analysis or advanced calculus course.

Desired Learning Outcomes:

Students will assimilate the definitions and basic concepts of measure theory, Lebesgue integration, L_p spaces, Banach spaces and Hilbert spaces. Students will learn the statements of a number of fundamental theorems, and will study their proofs. Students will be doing homework problems which will involve some computations but mostly the proving of various facts. The majority of the assessment will consist of written exams similar to the homework problems.

Homework/Exams/Grades:

There will be regularly assigned homework. Most of it will not be handed in, however we will discuss homework problems in class and I will post some solutions. A few homework problems during the semester will be handed in. There will be a short quiz around the end of the second or third week, two exams later in the semester, one of which may be a take-home exam, and there will be a final exam on the college scheduled final exam day. Your course grade will be based on the quiz and exams, according to the following rubric: Quiz - 15%, Exams - 25% each, Final - 25%, occasional homework and class participation - 10%.

The Homework Assignments

• Assignment One
  Ex. 2A: pg. 23/2,7,10,11
  Ex. 2B: pg. 38/3,6,10,12,22,28.
  


• Assignment Two. Do all the problems, but only the starred problems are to be handed in, due October 19:
  Ex. 2D: pg. 60/14,15,17*,19*,22
  Ex. 2E/4,5
  Ex. 3A: pg. 84/3,6,12,13*,19*
  Ex. 3B: pg. 99/5,7,10,12*,14*
  


• Assignment Three, but only the starred problems are to be handed in, due December 13, 2023
  Ex. 6C: pg. 170/4,7*,10, 12
  Ex. 7A, pg 199/4*,5*,14,17*
  Ex. 7B, pg 208/6,7,11*
  

Topics:

This course is an introduction to Real Analysis, taught at a fairly abstract and conceptual level, with an emphasis on definitions, theorems, and proofs. The students will be doing proofs in the homework, and on exams. Here is a list of topics we hope to cover, roughly keyed to the table of contents of Axler's Book.

• Chapter Two: Measures
• Chapter Three: Lebesgue Integration
• Chapter Six: Banach Spaces
• Chapter Seven: L_p Spaces
• Chapters Eight-Eleven: Hilbert spaces and Fourier Analysis