Functional analysis and harmonic analysis

Mathematical sciences LTH course on advanced level autumn 2017

New time for the next lecture!

Next lecture will be Monday September 18 at 8-10 in MH 333 and not on Friday.

Exercises

Exercises are discussed 13.15 on Friday September 22 in MH 333.

Course literature

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

Lecture notes 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.
4 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.
8 Compactness, continuity.
1 Operator theory 8.1.
2 Spectral theory 8.3.
4 Selfadjointness 8.4, Repetition.

Exercises

Repetition questions

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

Time and place

Tuseday 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 apperntly different fields.

The course is intended to be taken together with the course Partial differential Equations with Distribution Theory, FMA 250, (using the same course literature), but can without problems be read as a standalone course.

Some facts about the course

Lecturer:
Pelle Pettersson
Time:
Lp 1 and 2 autumn 2017
Duration:
Lectures 28 h and problem sessions 28 h.
Code:
FMAN80 (FMA260Fny).
Points:
7.5.
Literature:
Renardy and Rogers: An introduction to Partial Differential equations, Second edition, Springer.