Advanced Real Analysis I, SF2743

Starting day: Tuesday, Sept. 3, 13.15-15.00, at room 3733, Building of the Dept. of Mathematics.

Schedule: Tuesdays 13.15-15.00 room 3733.

Teachers: Henrik Shahgholian and Annemarie Luger.

Language: English

Goal:

To learn basics of advanced real analysis, such as the notion of measure, Lebesgue integral, and certain convergence aspects. Basics of functional analysis such as Banach spaces and linear operators will be introduced. These concepts will then be looked at from an application point of view, that may includesome of these topics: Fourier analysis, Ergodic theory, Probability theory, Sobolev spaces, Partial Differential equations. After completing the course the students are expected to:

i) Be familiar with concepts in measure, integration, and operator theory,

ii) Be able to state and prove basic theorems in measure and operator theory,

iii) Be able to use methods from above within applications.

Topics:

Measure and Integration (6 lectures):

Basics of measure theory:

Integration on measure spaces (Lebesgue integral):

Convergence theorems:

Product measures, and Fubini's theorem.

Section 1.1-1.6 (1.7) ; 2.1-2.11, 2.14-2.16

Functional Analysis (5 lectures):

Introduction to functional analysis

Banach spaces, including Lp-spaces

Basic theorems on linear operators and linear functionals

Section: 3.1-3.7 (3.8) 4.1-4.6 (4.7) 4.8 (4.9)

Applications (3 lectures): These may be chosen from:

Fourier analysis,

Ergodic theory,

Probability theory,

Sobolev spaces,

Partial Differential Equations.

This year’s topics:

1)Probability: Measure theory, filtering and conditional expectation
2)Ergodic Theory
3)Fourier Analysis

Prerequisites: Analys grundkurs SF2713, or equivalent.

Literature: A. Friedman, Foundations of Modern Analysis, Dover 1982

Examination:

The examination consists of several parts:

Home works, Midterm exam, Final, and possible oral exam.

Home works:

There will be 2 series of home works, that upon passing gives you the right to drop 1 question, for each home work, from the exam

The exam will consist of a FINAL which has two parts:

Part A)

General knowledge is required, and certain questions/problems will be suggested to read, as well as some simple cases of theorems.

Part B)

This part will require a deeper understanding of the topic, as well as proving/handling theorems and examples/exercises of more complexity.

Grade PASS (E)

You need to take Part A of the final exam.

Grades A-E

You need to have grade E, and in addition to take points from Part B.

For Graduate students in Mathematics the requirement is to have at least grade B to consider this course as PASS.

-----------------------------------------------------------------------------------------------------------

Course Schedule:

Glossary

Follow the link