# Real Analysis

### Syllabus

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

• The final grade will be computed as a grade point average with the following weights:

 Homework: 30% Midterm Exam: 30% Final Exam: 40%

• Each of the individual scores will be converted to Jacobs grade points before the overall weighted grade point average is computed.

### 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