Functional analysis and harmonic analysis

Mathematical sciences LTH course on advanced level autumn 2020

Start

The course starts at 10.15 on Tuesday September 1 in MH 333.

(Due to an expected limited set of participants, the course is planned to be held in a usual way in a classroom.)

Welcome!

Course literature

Renardy and Rogers is available on the net from computers connected to the LU net.

Since the course book is quite theoretical complete Lecture notes covering the material will be handed out during the course together with exercises.

Secondary literature

In Mathematical structures there are background material to parts of the course.

Program

Week Content. Section in RR
1 Introduction. 6.1.
1 Banach and Hilbert spaces. 6.1.
2 Hilbert space geometry. 6.1.3, 6.3.1, 6.3.2.
3 Basis. 6.2.1, 6.2.2.
3 Weierstrass approximation. 6.2.3, 2.3.3.
4 Dual spaces, Hahn-Banach. 6.3.1, 6.3.2, 6.3.3.
5 Weak convergence, uniform boundedness. 6.3.4, 6.3.5.
5 Sobolev spaces. 7.1.
6 Fourier transform, Sobolev imbedding theorem 7.2.2.
7 Sobolev imbedding theorem 7.2.3.
1 Compactness, continuity.
2 Operator theory 8.1.
4 Spectral theory 8.3.
5 Selfadjointness 8.4, Repetition.

Exercises

Sheets with exercises will be handed out during the course.

The exercises are discussed during the exercise sessions every second Friday.

Repetition questions

To recieve a pass in the oral exam one should do well on the following repetition questions.

Time and place

Tuesday 10-12 and Friday 13-15 in MH 333 (see timeedit).

Supplementary reading

The following are three books on functional analysis which are free on the net.

Aim

Functional analysis and harmonic analysis are fundamental tools in many important areas of abstract and applied mathematics as well as mathematical statistics and numerical analysis. The aim of the course is to convey knowledge about basic concepts and methods, and to give the ability, both to follow discussions where these are used and to independenty solve mathematical problens which arise in the applications. An important goal of the course is also to develop a power of abstraction which makes it easier to see analogies between problems from apperently different fields.

Some facts about the course

Lecturer:
Pelle Pettersson
Time:
Lp 1 and 2 autumn 2020
Duration:
Lectures 28 h and problem sessions 28 h.
Code:
FMAN80 (FMAN80F).
Points:
7.5.
Literature:
Renardy and Rogers: An introduction to Partial Differential equations, Second edition, Springer.