Math 8601-8602: Fall 2006/Spring 2007

Math 8601-8602: Real Analysis

Homework 1 , Homework 2 , Homework 3
Midterm 1 - solutions
Homework 4 , Homework 5 , Homework 6
Midterm 2 - solutions
Homework 7 , Homework 8 , Homework 9 , Homework 10 , Homework 11
Midterm 3 - solutions
Homework 12

Homework 13 , Homework 14 , Homework 15 , Homework 16 , Homework 17
Midterm 4 - solutions
Homework 18 , Homework 19 , Homework 20 , Homework 21 , Homework 22 , Homework 23
Midterm 5 - solutions
Homework 24 , Homework 25
Final Exam

The CourseLib page
Lecture notes by Walter Rusin
A note on separation theorems by Lu Li

Instructor. Marta Lewicka (office hours in VinH 328, Monday 4:30-5:15pm, Friday 2:30-3:15pm)
Recommended texts. The course will be self-contained, however the following books are recommended:
- Lang: Real analysis
- Royden: Real analysis
- Rudin: Functional analysis
- Brezis: Analyse fonctionelle - Theorie et applications
- Lieb and Loss: Analysis
- Ambrosio, Fusco and Pallara: Functions of bounded variation and free discontinuity problems
- Oxtoby: Measure and category
- Gelbaum and Olmsted: Counterexamples in analysis
Prerequisites. The course is intended for graduate students; undergraduates require Math 5615-5616 and the permission of the instructor. Also, you should read first two chapters of Royden's book (set theory and the real number system) yourself; we will use it as necessary in the course and will assume you are comfortable with this material. If not, please contact the instructor in the beginning of the course.
Grades for 8601. Grades will be based on homework (40%) and three midterms (60%). The grade distributions
Grades for 8602. Grades will be based on homework (30%), two midterms (30%) and the final exam (40%). Incompletes and make-up exams will almost never be given, and only for cases of extreme personal tragedy. The grade distributions
Homework. Homework will be assigned each Friday, and due the following Friday at the beginning of class. Late homework will not be accepted. The solution of each exercise will be evaluated in the scale 0-5 points, taking into account the correctness, clarity and neatness of presentation. You may (and should) collaborate and discuss the problems with each other but should write up solutions independently.
Topics. This sequence of courses, will provide a rigorous introduction to the following subjects:
I. Differentiation: I.1. Banach spaces and linear operators, I.2. The Hahn-Banach Theorem, I.3. Extensors and extensions, I.4. Differential calculus in Banach spaces, I.5. The inverse function theorem, I.6. Higher derivatives and the Taylor's formula.
II. Integration: II.1. Abstract measures, Caratheodory's theorem, II.2. Measurable functions, II.3. Lebesgue measure, Vitali's and Sierpinski's sets, II.4. Integral with respect to a measure, passing to the limit under the integral, II.5. Lebesgue integral vs. Riemann integral, II.6. Product measures and the Fubini-Tonelli theorem, II.7. The change of variable formula, II.8. The Egoroff and Luzin theorems, II.9. Differentiability of Lipschitz functions, Rademacher's theorem, II.10. Baire category.
III. Functional analysis and main functional spaces: III.1. Banach-Steinhaus theorems, III.2. Banach theorem of open mapping, closed graph, III.3. Weak (*) topologies and weak (*) convergence, Mazur's theorem, the separation theorems (geometrical form of Hahn-Banach), III.4. Compactness and sequential compactness, the theorems of Banach-Alaoglu, Kakutani, Goldstine, III.5. Reflexivity, separability, uniform convexity of Banach spaces, III.6. Hilbert spaces, Riesz-Frechet theorem, projection theorem, Lax-Milgram theorem, Hilbert sums and Hilbert basis, Bessel-Parseval identity.
IV. Main functional spaces: IV.1. The Lebesgue spaces L^p, Holder inequality, separability, reflexivity, Riesz representation theorem, IV.2. Convolution and mollification, density results, IV.3. Riesz-Frechet-Kolmogorof theorem, IV.4. Jenssen's inequality, IV.5. Lebesgue points, IV.6. Sobolev spaces W^{k,p}, separability, reflexivity, examples, IV.7. Friedrichs' lemma, Meyers-Serrin theorem (H=W), IV.8. The extension operator, IV.9. Embedding theorems of: Sobolev-Gagliardo-Nirenberg and Morrey for W^{1,p}(\Omega), IV.10. Rellich-Kondrachov theorem, IV.11. Trace operator and the space W_0^{1,p}(\Omega), IV.12. Poincare inequalities.
V. Measure theory and BV functions: V.1. Nonnegative Radon measures, Vitali-Besicovich theorem, differentiation of Radon measures, V.2. Radon-Nikodym theorem, Lebesgue deconposition theorem, V.3. Riesz representation theorem, V.4. Vector measures, variation of a measure, Jordan, Hahn, polar decompositions, V.5. Functions of bounded variation, perimeter of a set, V.6. Helly's compactness theorem, V.7. The 1-dimensional case.

Calendar.

6 Sept (W): First class
27 Sept (W): First Midterm
20 Oct (F): Second Midterm
24 Nov (F): Thanksgiving - no class
27 Nov (M): Third Midterm
13 Dec (W): Last class

17 Jan (W): First class
23 Feb (F) : First Midterm
12 March - 16 March (M, W, F): Spring break - no classes
20 Apr (F) : Second Midterm
4 May (F): Last class
10 May (Th) 8:00am - 10:00am: Final exam in VinH 6