COURSE: MATH 5220-001

TIME: 11:15-12:35 TR, PLACE: Room 210 Yoakley Hall

INSTRUCTOR: Dr. Robert Gardner, OFFICE: Room 308F of Gilbreath Hall

OFFICE HOURS: 12:40-1:30 TR PHONE: 439-6979 (Math Office 439-4349)

E-MAIL: gardnerr@etsu.edu
WEBPAGE: http://faculty.etsu.edu/gardnerr/gardner.htm

TEXT: Real Analysis, Fourth Edition, by H.L. Royden and P.M. Fitzpatrick, Prentice Hall (2010). Also, Chapter 5 of Real Analysis with an Introduction to Wavelets and Applications, by D. Hong, J. Wang, and R. Gardner, Academic Press/Elsevier Press (2005) will be covered.

ABOUT THE COURSE: We will build on the results of Real Analysis 1. We'll finish Lebesgue integration. We will cover, to some extent, Banach spaces and Hilbert spaces, though these are topics more appropriately covered in a functional analysis class (which I am currently developing and will offer summer 2013). Depending on class interest, we will look at topological spaces and metric spaces. Time permitting, I want to cover some of the topics on general measure and integration such as signed measures, product measures, and the Fubini-Tonelli results. We will use the Royden text for most topics, but you will be given a handout for Hilbert spaces from Real Analysis with an Introduction to Wavelets and Applications by D. Hong, J. Wang, and R. Gardner (yeah, that's right!). The fourth edition of Real Analysis states on page x that "The general theory of measure and integration was born in the early twentieth century. It is now an indispensable ingredient in remarkably diverse areas of mathematics, including probability theory, partial differential equation, functional analysis, harmonic analysis, and dynamical systems. Indeed, it has become a unifying concept."

ONLINE NOTES: I will try to have online notes in PDF form available for each section we cover. The notes will include definitions, some motivational comments, and statements of lemmas, theorems, and corollaries. The notes can be found at: http://faculty.etsu.edu/gardnerr/5210/notes1.htm and http://faculty.etsu.edu/gardnerr/5210/notes3.htm

GRADING: Homework will be assigned on a regular basis (weekly) and your grade on the homework will determine your grade for the course. Grades will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate.

A NOTE ABOUT HOMEWORK: While I suspect that you may work with each other on the homework problems (in fact, I encourage you to), I expect that the work you turn in is your own and that you understand it. If I get homework from two (or more) of you that is virtually identical, then neither of you will get any credit. Some of the homework problems are fairly standard for this class, and you may find proofs online. However, the online proofs may not be done with the notation, definitions, and specific methods which we are developing and, therefore, are not acceptable for this class. You are expected to give all details and document all claims on the homework!!!

TENTATIVE OUTLINE:
We will cover topics from this list:
Chapter 4: Lebesgue Integration (4.4 to 4.6).
Lebesgue Dominated Convergence Theorem, general Lebesgue integral, uniform integrability, Vitali Convergence Theorem.
Chapter 5: Lebesgue Integration: Further Topics.
General Vitali Convergence Theorem, convergence in measure, characterizations of Riemann and Lebesgue integrability.
Chapter 6: Differentiation and Integration (partial).
Vitali Covering Theorem, bounded variation, absolute continuity, differentiation and integration, convex functions, Jensen's Inequality.
Chapter 7: The L^p Spaces: Completeness and Approximation.
L^p spaces, Minkowski and Holder Inequalities, convergence and completeness, Banach spaces, Riesz-Fischer Theorem, approximation, and separability.
Chapter 8: The L^p Spaces: Duality and Weak Convergence.
Bounded linear functionals, Riesz Representation Theorem, dual spaces, weak convergence.
Vector Spaces, Hilbert Spaces, and the L² Space (Hong, Wang and Gardner; partial).
Groups, fields, and vector spaces, inner product spaces, L², projections and Hilbert space isomorphisms, Banach spaces revisited.
Chapters 9 and 10: Metric Spaces (maybe).
Topological definitions, continuity, homeomorphisms, completeness, separable spaces, compact spaces, Baire category.
Chapters 11 and 12: Topological Spaces (maybe).
Open/closed, continuous, bases, separation axioms, connectedness, compactness, product topology.
Chapters 17 and 18: General Measure (maybe).
Signed measure, Caratheodory measure, outer measures, Caratheodory-Hahn Theorem.
Chapter 20: Particular Measures (partial).
Product measures, multiple integrals, theorems of Fubini and Tonelli.

IMPORTANT DATES: (see http://www.etsu.edu/etsu/academicdates.aspx for the official ETSU calendar):

• Monday, January 21 = Martin Luther King, Jr. Holiday.
• Wednesday, January 30 = Last day for 75% refund of fees.
• Wednesday, January 30 = Last day to drop without a grade of "W."
• Wednesday, February 13 = Last day for 25% refund of fees.
• Monday through Friday, March 11 to March 16 = Spring Break Holiday.
• Wednesday, March 13 = Last day to drop without dean's permission.
• Wednesday, May 1 = Last day to withdraw from the university.
• Friday, May 3 = Last day of class.

OTHER RESOURCES. The following were mentioned in class:

1. To access the Mathematical Reviews: Go to the Sherrod Library online catalog. Click the "Title" tab and enter "Mathematical Reviews." Select "MathSciNet [Electronic Resource]" and follow the links. You will be asked to enter your user ID and password (the same you use for your ETSU e-mail). You are then redirected to MathSciNet and can freely use it and even download PDF versions of some of the papers you find! Of course, this protocol will work for any electronic journal available through the Sherrod Library.

Return to Bob Gardner's home page
Last updated: April 23, 2013.