Instructor. Dr. Marta Lewicka (office hours in Thack
620, Monday, Wednesday 3:00-4:00pm) Grader. Jared Burns
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 (50%) and two
midterms (25% + 25%).
Incompletes and make-up exams will almost never be given,
and only for cases of extreme personal tragedy.
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 collaborate and discuss the problems with each other
but should write up solutions independently.
Topics. This course will provide a rigorous
introduction to the basic subjects lying at the heart of
modern Mathematical Analysis, which are: Measure and Integration,
Basic Functional Analysis, Lebesgue Spaces and Sobolev Spaces.
These topics are also of the 'daily bread' of the contemporary theory
of Partial Differential Equations, Differential Geometry, Geometrical
Analysis, Harmonic Analysis, Engineering, Mechanics and Physics.
I. Measure and Integration:
I.1. Abstract measures, Caratheodory's theorem,
I.2. Measurable functions,
I.3. Lebesgue measure, Vitali's set (mention Sierpinski's set),
I.4. Integral with respect to a measure, passing to the limit under the integral,
I.5. Lebesgue integral vs. Riemann integral,
I.6. Product measures and the Fubini-Tonelli theorem,
I.7. The change of variable formula,
I.8. The Egoroff and Luzin theorems,
I.9. Baire category and applications.
II. Functional Analysis and Main Functional Spaces:
II.1. Banach spaces and linear operators,
II.2. The Hahn-Banach Theorem,
II.3. Banach-Steinhaus theorems,
II.4. Banach theorem of open mapping, closed graph,
II.5. Weak (*) topologies and weak (*) convergence, Mazur's theorem,
the separation theorems (geometrical form of Hahn-Banach),
II.6. Compactness and sequential compactness,
the theorems of Banach-Alaoglu, Kakutani, Goldstine,
II.7. Reflexivity, separability, uniform convexity of Banach spaces.
III. Main Functional Spaces: L^p and Sobolev Spaces
III.1. The Lebesgue spaces L^p, Holder inequality, separability, reflexivity,
Riesz representation theorem,
III.2. Convolution and mollification, density results,
III.3. Riesz-Frechet-Kolmogorof theorem,
III.4. Jenssen's inequality,
III.5. Lebesgue points,
III.6. Sobolev spaces W^{k,p}, separability, reflexivity, examples,
III.7. Friedrichs' lemma, Meyers-Serrin theorem (H=W),
III.8. Embedding theorems of: Sobolev-Gagliardo-Nirenberg and Morrey for W^{1,p}(R^n).
Calendar.
29 Aug (M): First class
5 Sep (M): Labor Day - no class
10 Oct (M): Fall break - no class
11 Oct (Tue): Class from Monday
31 Oct (M): First Midterm
23 Nov (W), 25 Nov (F): Thanksgiving - no class
12 Dec (M): Second Midterm