## Math 6320, Introduction to Real Analysis 2

## Semester 2, 2019-20

**Course Lecturer: **
#### Dr. Judith Packer, Dept. of Mathematics

*Tel: (303) 492-6979*

Office: Math 227

Email: packer@euclid.colorado.edu

URL: http://math.colorado.edu/~packer

**Course Information:**

This course is meant to continue the study of analysis of real-valued functions of one or several variables,
with an emphasis on Lebesgue measure
and Lebesgue integration on the real line and **R^n**.
Topics to be covered include:

Locally compact Hausdorff spaces, the Stone-Weierstrass Theorem, the Arzela-Ascoli Theorem;
Elements of functional analysis, including normed
vector spaces, linear functionals, the Baire Category Theorem;
Measure and integration: general theory, signed measures - the Hahn and Jordan decomposition
theorem, outer measures and Caratheodory's extension theorem,
the Radon-Nikodym Theorem; General **L^p ** spaces and duality;
Product measures, the Fubini and Tonelli Theorems, cumulative distribution theorems and
Borel measures on ** R**;
Radon measures: positive linear functionals on **C_C(X)** and the Riesz representation theorems,
regularity of measures, the dual of $C_0(X),$ products of Radon measures;
Elements of Fourier analysis:
convolutions, the Fourier transform, summation of Fourier integrals and series
and appplications of Hilbert space theory, pointwise convergence of Fourier series, Fourier analysis of measures.

**Prerequisite:**

Math 6310, or instructor consent.

**Course Text:**

We will use as a primary text the book
"Real Analysis, 4th Edition, by Halsey Royden and Patrick Fitzpatrick, Pearson, 2010, covering much of Chapters 10, 12, 16,
17, 18, 19, and 20.

Fitzpatrick has an errata page for this textbook at
http://www2.math.umd.edu/~pmf/docs/Real%20Analysis.pdf

**Assessment:**
- Homework Assignments (every two weeks or so): 30 %
- Midterm test 1 in class - Friday, Feb. 21, 2020, 1 p.m - 1:55 p.m. : 20 %
- Sample first midterm , from Spring 2016.
- Solutions to sample first midterm .
- Solutions to first midterm given Feb. 21, 2020
.
- Midterm test 2 take-home: given out Monday, March 30, 2020, 1:50 p.m., due Monday April 6, 2020, 1 p.m. : 20 %
- Sample final exam , from Spring 2019.
- Final exam, Monday, May 4, 2020 in class , 4:30 p.m. - 7 p.m. : 30 % .
- Click HERE for a detailed syllabus of the course, including all university
policies and procedures.

**Lecture Hours and Venue: **

MWF 1 p.m.-1:50 p.m. KCEN N252.

**Office Hours: **

MWF 11-12, and by appointment.

** Homework: **

- Assignment 1 , due Wednesday, Jan. 29, 2020.
- Some solutions to Assignment 1.
- Assignment 2 , due Friday, February 14, 2020.
- Some solutions to Assignment 2.
- Assignment 3 , due Wednesday, March 11, 2020.
- Assignment 4 , due Friday, March 20, 2020 (e-mail me scanned copies of your solutions).
- Some solutions to Assignments 3 and 4.
- Assignment 5 , due Wednesday, April 22, 2020 (upload scanned pdf files of your solutions to Canvas).
- Some solutions to Assignment 5.

**Some Important Names associated with Real Analysis :**

Back to the home page of Judith A. Packer

Last modified December 20, 2019.