Fall Semester 2008

Real Analysis

Syllabus

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Summary:
This course is an introduction to measures, integration, and elements from from functional analysis and the theory of function spaces. While abstract measure theory is covered, the focus of this course is on results and techniques which can be proved in the more concrete setting of Lebesgue integration. The course is suitable for undergraduate students who have taken Analysis I, Analysis II, and Linear Algebra I; it should also be taken by incoming graduate students. The graduate course in Functional Analysis will continue the topics and also provide a more abstract framework for several of the topics presented here.

Contact Information:
Instructor:Marcel Oliver
Email:m.oliver@jacobs-university.de
Phone:200-3212
Office hours:  Mo, Fr 11:15 in Research I, 107
Grader:Mahmut Çalik
Email:m.calik@jacobs-university.de

Time and Place:
Lectures:  We 11:15, Fr 9:45 in East Hall 3

Recommended Textbook:
E.H. Lieb, M. Loss: Analysis, second edition, AMS, 2001
G.B. Folland: Real Analysis, second edition, Wiley, 1999

Grading:

Class Schedule (subject to change!)

03/09/2008: Motivation, sigma-algebras, measures
08/09/2008: Construction of the Lebesgue measure I: From outer measure to measure
10/09/2008: Construction of the Lebesgue measure II: From pre-measure to outer measure; monotone class theorem, uniqueness of measures
15/09/2008: Measurable functions and integrals
17/09/2008: Monotone convergence, Fatou's lemma, dominated convergence
22/09/2008: Product measures, Fubini-Tonelli theorem
24/09/2008: Proof of the Fubini-Tonelli theorem; Layer cake representation
29/09/2008: Normed spaces, Lp-spaces, Jensen's inequality
01/10/2008: More inequalities: Young, Hölder, Minkowski, Hanner
06/10/2008: Completeness of Lp; Bounded linear functionals on Banach spaces, dual space, weak convergence
08/10/2008: uniform boundedness principle
13/10/2008: Dual of Lp
15/10/2008: Convolution, mollifiers, separability of Lp
20/10/2008: Midterm review
22/10/2008: Midterm Exam
27/10/2008: Midterm discussion
29/10/2008: Approximation by "really simple functions", separability of Lp
03/11/2008: Convolution, convergence of a mollifying sequence in Lp
05/11/2008: Banach-Alaoglu for Lp, Hilbert spaces (begin)
10/11/2008: Hilbert spaces
12/11/2008: Fourier Transform
17/11/2008: Fourier Transform
19/11/2008: Fourier series, pointwise convergence of Fourier series for continuous functions of bounded variation
24/11/2008: Stone-Weierstrass Theorem
26/11/2008: Distributions
28/11/2008: Distributions
03/12/2008: Review for final exam
18/12/2008: Final Exam, 16:00-18:00 in Research II Lecture Hall



Last modified: 2008/11/27
This page: http://math.jacobs-university.de/oliver/teaching/jacobs/fall2008/math362/index.html
Marcel Oliver (m.oliver@jacobs-university.de)