## MATH 746 Theory of Functions of Real Variable I Fall 2020

Rob Thompson Hunter College

Tuesday--Thursday 7:00-8:15pm Room: Online

August 26-December 20, 2020

**e-mail:**
robert.thompson@hunter.cuny.edu

**Office:** http://zoom.us/my/profthompson
**Hours: Tuesday and Thursday, 6:00-7:00.**

### Please Note: This course will be held online.
We will have synchronous, live, online classes, using
Zoom, during the regularly scheduled time slot for the course. The
office hours will also be online. To access my Zoom meeting space go
to http://zoom.us/my/profthompson in your browser and you will be
prompted to go to your zoom app. If you do not have a Zoom app
installed you will have to do that first. You do not necessarily need
to have an account with Zoom to install and use the app. If you go into the Zoom app directly just go to
profthompson.

### We will have an exam Thursday, 10/22. Topics covered will be Chapter Two, Sections A-D, and Chapter Three, Section 3A.

Here are some Homework solutions/hints.
### We will have a quiz Tuesday, 9/22. The topics will be on measure theory.

Specifically,
- The definition of a sigma algebra, Borel set, outer measure, Lebesgue measureable set, Lebesgue measure.
- The quiz won't have anything about measurable functions
Here are solutions to the quiz.

### Here are some practice problems. I will post more.:

**My Office:**

My office hours for the fall term will for the most part be **Tuesday and Thursday 6:00-7:00 pm**. However there will be a couple of days when
I have to change this, but I will announce that. You don't need to make an appointment for office hours, you can
just drop in. The Zoom address is **profthompson**.

**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 book of this publishing quality.
**Not required but a good reference:**Real Analysis by H. L. Royden.

**Prerequisites:**

MATH 351 or the equivalent.

**Desired Learning Outcomes:**

The student will assimilate the
definitions and basic concepts of measure theory, Lebesgue integration, L_p spaces, and Banach spaces.
The student will learn the statements of a number of
fundamental theorems, and will study their proofs. The student 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 will probably be a take-home. 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%.

### Recorded Classes

** The Homework Assignments**

- Assignment One, not to be handed in, but should be done by next Tuesday September 15.

Ex. 2A: pg. 23/2,7,10,11

Ex. 2B: pg. 38/3,6,10,12 (this is the one I mentioned in class),22,28.

- Assignment Two, to be
**handed in**, due October 20:

2D/14,15,17,19,22

2E/4,5

3A/3,6,12,13,19

- Assignment Three

3B, page 99: 5,7,10,12,14

6C, page 170:4, 7, 10, 12

### Instructions for submitting the homework:

**The homework must be in the form of PDF file**
- Go to the BlackBoard page for this class
- On the left hand side there is a menu, select "Upload HW here"
- Select the Problem Set you are submitting (e.g. Problem Set 1)
- Ignore the item that says "Point Possible"
- Select the"Browse my computer" button
- Find your PDF file which is your homework
- Upload it

**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 One: Riemann Integration
- Chapter Two: Measures
- Chapter Three: Lebesgue Integration
- Chapter Four: Differentiation
- Chapter Six: Banach Spaces
- Chapter Seven: L_p Spaces
- CHapters Eight-Eleven: Hilbert spaces and Fourier Analysis