# Function spaces, fall 2008

### Lecturer

### Type

Advanced studies.

### Lectures

Weeks 37-42 and 44-50, Wednesday 14-16 in room B322 and Thursday 14-16 in room B321.

### Lecture notes

The lecture notes for 1. period may be found below as pdf files of scanned pages.

Lectures 1 and 2 (week 37)

Lectures 3 and 4 (week 38)

Lectures 5 and 6 (week 39)

Lectures 7 and 8 (week 40)

Lectures 9 and 10 (week 41)

Lectures 11 and 12 (week 42)

The lecture notes for 2. period are here in pdf form.

Lecture notes (final version, updated 29.01.2009)

### Exercises

It is possible to substitute one midterm by doing exercises, and in this case one can get a maximum of 4 points from each exercise set (thus it is enough to do 4 problems in each set). The exercises can be returned to Lauri Ylinen (room B329).

Exercises 1 (18.09.2008, due on 25.09.2008)

Exercises 2 (02.10.2008, due on 09.10.2008)

Exercises 3 (16.10.2008, due on 30.10.2008)

Exercises 4 (06.11.2008, due on 13.11.2008)

Exercises 5 (20.11.2008, due on 27.11.2008)

Exercises 6 (05.12.2008, due on 19.12.2008)

### Course description

In mathematical analysis one deals with functions which are differentiable (such as continuously differentiable) or integrable (such as square integrable or L^p). It is often natural to combine the smoothness and integrability requirements, which leads one to introduce various spaces of functions.

This course will give a systematic introduction to those function spaces which are most commonly encountered in analysis. This will include Hölder, Lipschitz, Sobolev, Besov, Triebel-Lizorkin, and Zygmund type spaces. We will try to highlight typical uses of these spaces, and will also give an account of interpolation theory which is an important tool in their study.

The program for the course is as follows:

- Sobolev spaces of integer order
- Interpolation theory
- Fractional Sobolev spaces, Besov and Triebel-Lizorkin spaces

### Prerequisites

Funktionaalianalyysin peruskurssi, Reaalianalyysi 1

### Credits

10 sp