 Henri Lebesgue, 1875-1941 |
 Stefan Banach, 1892-1945 |
 David Hilbert, 1862-1943 |
Henri Lebesgue, 1875-1941 |
Stefan Banach, 1892-1945 |
David Hilbert, 1862-1943 |
COURSE: MATH 5220-001
TIME: 9:45-11:05 TR, PLACE: Room 231 Lamb Hall
INSTRUCTOR: Dr. Robert Gardner, OFFICE: Room 308F of Gilbreath Hall
OFFICE HOURS: 12:20-1:30 MWF 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.
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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!). 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."
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} (4.3 to 4.6)
Fatou's Lemma, Monotone Convergence Theorem, Lebesgue Dominated Convergence Theorem, general Lebesgue
integral, uniform integrability.
Chapter 5: Lebesgue Integration: Further Topics
Vitali Convergence Theorem, convergence in measure, characterizations of Riemann and Lebesgue integrability.
Chapter 6: Differentiation and Integration
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 H\"{o}lder 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 $L^2$ Space (Hong, Wang and Gardner)
Groups, fields, and vector spaces, inner product spaces, $L^2$, projections and Hilbert space isomorphisms, Banach spaces revisited.
Chapters 11 and 12: Topological Spaces
Open/closed, continuous, bases, separation axioms, connectedness, compactness, product topology.
Chapters 9 and 10: Metric Spaces
Topological definitions, continuity, homeomorphisms, completeness, separable spaces, compact spaces, Baire category.
Other Topics
More Hilbert spaces (Chapter 16), more measure (Chapter 17), product measure (20.1).
IMPORTANT DATES:
OTHER RESOURCES. The following were mentioned in class:
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Last updated: February 17, 2011.