Instructor:
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Jackie Shen,
VinH 539, (612) 625-3570; Office
Hours: MW 1:30-2:20pm.
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Prerequisite:
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Math
5615H Honors: Introduction
to Analysis. |
Topics:
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As specified in the School's
Qualifying
Exam Syllabi:
- Basic
Theory:
continuity, Riemann-Stieltjes integrals, bounded variation,
equicontinuity; Ascoli-Arzela, Stone-Weierstrass, Baire category
- Lebesgue
Integration Theory: outer measure, measurable sets and
functions,
integration;
Theorems: Egorov,
Lusin, convergence, Fubini, and Tonelli
- Differentiation
Theory: maximal functions, Lebesgue differentiation theorem,
Vitali's covering lemma, absolute continuity, monotonicity, convexity
- Abstract
Integration Theory: Borel measures, Caratheodory-Hahn
extension, convergence theorems, Hahn decomposition,
Radon-Nikodym theorem
- Harmonic/Functional
Analysis (Intro): Lp-spaces, convolution and
mollification, Hilbert spaces, orthonormal sets and Fourier
series, linear functionals, Plancherel Theorem
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Special Interest:
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If time permits, I also attempt
to touch on some favorite
topics
like: Distribution
theory, Sobolev functions, general BV theory,
martingale theory in probability, as well as some applications
of Real Analysis in modern Information and Pattern Theory.
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Ref. Books:
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- G. B. Folland: Real Analysis, John Wiley & Sons, Inc. 1999.
(Detailed, comprehensive, but def-thm-proof style could lose you)
- R. L.Wheeden and A. Zygmund: Measure and Integral,
Marcel Dekker, New York, 1977. (Used in preceding years by Prof. Max Jodeit,
and strongly recommended by Prof. Max Jodeit)
- W. Rudin, Real and Complex Analysis, 3rd
Ed., McGraw Hill, New York, 1987. (One of the most popular)
- For graduate courses,
I never strictly follow any particular book. Attending lectures and
taking notes are important for you.
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Course Load:
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There will be weekly homeworks
(graded by a paper grader); one midterm (Friday, March 11,
during
the normal lecture time),
and one final ( 12:20pm-1:20pm,
Wednesday, May 4).
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Grading:
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Homework
30%; Midterm:
30%; Final Exam: 40%
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A Note:
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Among the 29 registered graduate
students, many are from Econ, Stat, EECS, or other
depts.
On the other hand, for math graduate students, this is a core course
often oriented towards qualifying exams. So I will try my best
for such a mixed audience.
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Holidays
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Spring Break: March 14-March 18, 2005
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