COURSE: MATH 5220-001

TIME: 2:15-3:35 TR, PLACE: Room 327 Rogers-Stout Hall

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

OFFICE HOURS: 11:15-12:00 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). We will use James R. Munkres' Topology, 2nd edition, Prentice Hall (2000) as a supplemental text.

ABOUT THE COURSE: We will build on the results of Real Analysis 1. We will cover, to some extent, Banach spaces and Hilbert spaces, though these are topics more appropriately covered in a functional analysis class (which will be offered during summer 2015). We will look at topological 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. 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: Online notes in PDF form are 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
• http://faculty.etsu.edu/gardnerr/5210/notes2.htm
• 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 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.
Chapters 11 and 12: Topological Spaces.
Open/closed, continuous, bases, separation axioms, connectedness, compactness, product topology.
Chapter 9 of Munkres Topology: The Fundamental Group (partial).
Homotopy of paths, covering spaces, the fundamental group of Sⁿ, the fundamental group of some surfaces.
Chapters 17 and 18: General Measure.
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; accessed 11/9/2014):

• Monday, January 26 = Last day to change to/from audit grade
• Monday, January 26 = Last day for to register or add without departmental permit.
• Monday, February 2 = Last day to drop without a grade of "W."
• Monday through Friday, March 9 to March 13 = Spring Break Holiday.
• Tuesday, March 10 = Last day to drop without dean's permission.
• Tuesday, April 28 = Last day to withdraw from the university.
• Thursday, April 30 = Last day of class.

Functional Analysis: The department will offer "Introduction to Functional Analysis" (MATH 5740) during summer term 1, 2015. Dr. Bob will teach the class and we will cover linear operators, Banach spaces in more detail, Hilbert spaces in much more detail, linear operators, the Hahn-Banach Theorem, duality, and the spectrum of an operator. Details are online at: http://faculty.etsu.edu/gardnerr/Func/sillsummer15.htm.

1. List of typos in Royden and Fitzpatrick (accessed 11/9/2014).
2. We will cover part of James R. Munkres' Topology, 2nd edition, Prentice Hall (2000). You might find a PDF copy of this online.
3. A popular level book on the history of topology covering, in particular, the classification of surfaces and a discussion of the four color theorem, is Euler's Gem: The Polyhedron Formula and the Birth of Topology by David S. Richeson, Princeton University Press (2008). ETSU has a copy of this in the Sherrod Library (QA611.A3.R53 2008).
4. Another readable introduction to topology (at the popular level) is From Geometry to Topology by H. Graham Flegg, Crane, Russak & Company (1974). It includes the idea of transformational geometry and leads from geometry to topolgy in a transformational setting, and discusses surfaces by appealing to plane diagrams. ETSU has a copy of this in the Sherrod Library (QA.611.F48) and is is also in print by Dover publications.
5. 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.

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Last updated: April 29, 2015.