**Math 541 (Measure Theory and Functional Analysis)
Spring 2007 **

**
**### Course Description

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:** 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:**
https://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% |

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