Math 541 (Measure Theory and Functional Analysis) Spring 2007

This course is a continuation of Math 540. It comes in three parts. In Part I we develop the theory of measure and integration in abstract spaces and then in greater depth in spaces with additional structure (topological or metric spaces). Part II is an introduction to the basics of functional analysis (Hilbert and Banach spaces, Hahn--Banach Theorem, Open Mapping and Closed Graph Theorems, Principle of Uniform Boundedness, weak topologies and Alaoglu's theorem). An important source of examples will be various function spaces defined over abstract measure spaces. In Part III we will give a crash course in the theory of Sobolev spaces and distributions.

Contents

• Instructor information
• Course information
• Homework and exams
• Grading Policy
• Tentative Syllabus

• Instructor: Prof. Jeremy Tyson
• Office Location: 330 Illini Hall
• Office Phone: 244-4132
• Email: tyson@math.uiuc.edu
• Office Hours: Mon 3-4, Wed 1-2, Thu 1-2

• Lecture Times: MonWedFri 11:00-11:50
• Lecture Location: 443 Altgeld Hall
• Course web site: http://www.math.uiuc.edu/~tyson/541s07.html.

• Textbook: Introduction to Modern Analysis by Shmuel Kantorovitz (Oxford Graduate Texts in Mathematics, #8, Oxford University Press, 2003).
  We will cover the material in chapters 1 through 6 as well as part of Application II.

• Homework: There will be homework assignments due two to three times per month. Problems will mostly be taken from the textbook, although I will assign some outside problems as well. I will grade selected problems from each assignment. We will discuss solutions to some of the homework problems during class. Each student in the class will be asked to present the solution to at least one problem over the course of the semester.

• Exams: There will be one midterm exam and a final exam (both take-home exams).

• Grading Policy: Grades will be computed according to the following percentages:

  Homework	30%
  Midterm Exam	30%
  Final	40%

1. Measure and integration, Lp spaces, Lebesgue-Radon-Nikodym theorem, outer measure, constructions of measures
2. Measure spaces with additional structure: measure and topology, measure and metric, Lebesgue differentiation theorem, Hausdorff measure
3. Hilbert and Banach spaces, linear functionals, dual spaces, Riesz Representation Theorem, examples, Hahn-Banach theorem, reflexivity
4. Topological vector spaces, weak topologies, Alaoglu theorem, applications
5. Operator theory: Principle of Uniform Boundedness, Closed Graph and Open Mapping theorems, category
6. Distributions and Sobolev spaces, fundamental solutions of linear PDE