Functional analysis and applications: Math 797FN

Meeting : TuTh 9:30--10:45    LGRT 1234

Instructor : Luc Rey-Bellet

 Office :   1423 J LGRT
 Phone :  545-6020
 E-Mail :  luc@math.umass.edu
 Office Hours :   Tu 11:00--12:00,   Th 3:00--4:00,   or by appointment.

Classnotesand homework: Some (partial) classnotes will be posted and regularly updated HERE

• Homework 1: Problems 1--10 in the lecture notes. Due on February 15.

• Homework 2: Problems 11--20 in the lecture notes. Due on March 7.

• Homework 3: Problems 21--30 in the lecture notes. Due on April 2.

References: I will not follow any textbook very closely. You are however very strongly encouraged to consult textbooks on your own. There are dozens of books on functional analysis with the authors taking many different point of view. The books selected below are all in the same spirit as the class (or vice-versa).

• Functional Analysis, Peter Lax, Wiley-Interscinece 2002. Very well-constructed, very complete, a pleasure to read.
  
  
• Perturbation Theory for Linear Operators, Tosio Kato. Classics in Mathematics, Springer, 1995 (reprint of 2nd edition). This classics contains a lot advanced material on spectral theory.
  
  
• Introductory Functional Analysis with Applications, Erwin Kreyszig. Wiley, 1989. Excellent introductory text.
  
  
• Principles of functional analysis, Martin Schechter. American Mathematical Society, 2001 (2nd edition). Another excellent introduction, very pedagocial
  
  
• Functional Analysis (Methods of Modern Mathematical Physics Vol. 1) , Michael Reed and Barry Simon. Academic Press 1981. All what you need to know.
  
  
• Functional analysis, Kosaku Yosida. A classic which contains many nice applications. Springer, 1996 (6th edition).
  
  

Syllabus: Functional analysis deals with the structure of infinite dimensional vector spaces and (mostly) linear on such spaces. Many such spaces are spaces of functions, hence the name functional analysis, but much of the theory will developed for abstract spaces (spaces with a norm or a scale product). We shall assume that the reader has taken Math 624 (or an equivalent course) and is familiar with the basic objects of functional analysis: Banach spaces and Hilbert spaces, linear functionals and duals, bounded linear operators. We will review these topics but at a rather brisk pace. Our main goal is to develop a series of tools instrumental in the applications of functional analysis to PDE's, probability, ergodic theory, etc... Among the topics covered in this class are

• The fundamental theorems of functional analysis: Hahn-Banach theorem, Baire theorem, Inverse mapping and closed graph theorems.
• Spectral theory I: Compact operators, Fredholm operators and applications
• Positive operators
• Semigroups
• Spectral theory II: The spectral theorem
• Unbounded operators
• Banach algebras

Grade: Homework will be assigned on a regular basis. Each student will pick a project of his choice. I expect a written project at the end of the semester and each student will give an oral presentation to the class.