REAL ANALYSIS 2 - Spring 2007

Henri Lebesgue, 1875-1941

Stefan Banach, 1892-1945

David Hilbert, 1862-1943

COURSE: MATH 5220-001

TIME: 11:30-12:50 MW PLACE: Room 124 Rogers-Stout Hall

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

OFFICE HOURS: Tuesday 9:00-10:00 and by appointment. PHONE: 439-6979 (Math Office 439-4349)

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

TEXT: Real Analysis, Third Edition, by H. L. Royden. Also, Chapter 5 of Real Analysis with an Introduction to Wavelets and Applications, by D. Hong, J. Wang, and R. Gardner will be covered.

ABOUT THE COURSE: We will build on the results of Real Analysis 1 and apply Lebesgue integration to Banach spaces and Hilbert spaces. We will explore topological spaces and maybe metric spaces. Depending on class interest, we will also address topics in "general measure and integration" such as signed measures, product measures, and 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!). If there is sufficient interest, we may also study the mathematical foundations of quantum mechanics from Chapter 7 of L. Debnath and P. Mikusinski's Introduction to Hilbert Spaces with Applications, 3rd edition.

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. 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.

TENTATIVE OUTLINE:
Chapter 6: The Classical Banach Spaces.
Lp spaces, Minkowski and Holder Inequalities, convergence and completeness, Banach spaces, Riesz-Fischer Theorem, approximation in Lp, bounded linear functionals, Riesz Representation Theorem.
Vector Spaces, Hilbert Spaces, and the L2 Space (Hong, Wang and Gardner)
Groups, fields, and vector spaces, inner product spaces, L2, projections and Hilbert space isomorphisms, Banach spaces revisited.
Chapter 8: Topological Spaces
Open/closed, continuous, bases, separation axioms, connectedness, product topology.
Chapter 7: Metric Spaces
Topological definitions, continuity, homeomorphisms, completeness, separable spaces, compact spaces, Baire category.
Other Topics
More Banach spaces (Chapter 10); measure spaces, signed measure (Chapter 11); outer measure, product measure (Chapter 12); quantum theory, Heisenberg's uncertainty principle, Schrodinger equation of motion (Debnath and Mikusinski).

IMPORTANT DATES:

Homework
Section
Problems
Due Date
Points
5.4
14a, 20a, Bonus: 20b
Wednesday 1/24
3+3+(6)=6+(6)
6.1
1, 3, 4
Wednesday 1/31
3+3+3=9
6.2
5b, 7b, Derive the L0 "norm"
Wednesday 2/14
3+3+3=9
6.3
11, 12, 13
Wednesday 2/28
3+3+3=9
6.5; Hilbert 5.1
22, 24; 4, 8, Bonus: 3
Friday 3/2
1+3+3+3+(6)=10+(6)
Hilbert 5.2; 5.3
3, 4; 4, 5, Bonus 5.2.6
Wednesday 4/4
3+3+3+1+(6)=10+(6)
Hilbert 5.4
6, 8, 10, Bonus 5.5.5
Monday 4/16
3+3+3+(3)=9+(3)
8.1
6, 8, 10
Wednesday 4/25
3+3+3=9
8.2, 8.3, 8.4
15; 19; 32, Bonus: Give example of a T1 space
and a sequence in the space with two different limits
Wednesday 5/2
4+4+3+(3)=11+(3)
TOTAL
-
-
82+(24)
The numbers in parentheses represent bonus problems.

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