University of Pittsburgh Math 2301-2302 : Graduate Analysis 1 and 2: Fall 2019, Spring 2020

Math 2301-2302: Graduate Real Analysis 1 and 2

Math 2301: Homework 1 , Homework 2 , Homework 3 , Homework 4 , Homework 5 , Homework 6 , Homework 7 , Homework 8 , Homework 9 , Homework 10 .

Math 2302: Homework 11 , Homework 12 , Homework 13 , Homework 14 , Homework 15 .

Instructor. Dr. Marta Lewicka (office hours in Thackeray Hall 408, Monday 3:20-4:20pm)
Recommended texts. The course will be self-contained, however the following books are recommended:
- Lang: Real analysis
- Royden: Real analysis
- Rudin: Functional analysis
- Brezis: Functional analysis, Sobolev spaces and Partial Differential Equations (also, you may read the shorter French edition: Analyse fonctionelle - Theorie et applications)
- Lieb and Loss: Analysis
- Oxtoby: Measure and category
- Gelbaum and Olmsted: Counterexamples in analysis
Prerequisites. The course is intended for graduate students; undergraduates require permission of the instructor (easily granted). Also, you should familiarize yourself with first two chapters of Royden's book (set theory and the real number system); 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. Grades will be based on homework (40%) and two midterms (30%+30%). Incompletes and make-up exams will almost never be given, and only for cases of extreme personal tragedy.
Homework. Homework will be assigned on certain Wednesdays and will be due the following Wednesday 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.
Topics. This course will provide a rigorous introduction to the basic subjects of modern Mathematical Analysis, which are: Differentiation, Measure and Integration, and Basic Functional Analysis. These topics are fundamental for: Partial Differential Equations, Differential Geometry, Geometrical Analysis, Harmonic Analysis, Engineering, Mechanics and Physics.

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's theorem.

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.

27 Aug (Mon): First class
30 Oct (Wed): First midterm
25 Nov -- 29 Nov: Thanksgiving break - no class
9 Dec (Mon): Second midterm

6 Jan (Mon): First class
20 Jan (Mon): Martin Luther King's day - no class
9 Mar - 11 Mar (Mon, Wed): Spring break - no classes
16 Mar : First midterm
22 Apr (Wed) : Second midterm