Fördjupningskurs till lineär funktionalanalys

**Valid from:** Autumn 2020**Decided by:** Professor Thomas Johansson**Date of establishment:** 2020-09-24

**Division:** Mathematics**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course code:** MATP45**Teaching language:** English

The main goal of the course is to give a presentation of relevant applications of the abstract principles of functional analysis to a large variety of problems in mathematical analysis.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to analyse problems in mathematical analysis using methods from functional analysis,
- be able to give examples of important applications of the abstract methods and principles of functional analysis,
- be able to give a detailed account of the theory behind methods described in the course,
- be able to give an account for research aspects within the subject and relate it to relevant problems within an independent work.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to critically and systematically integrate knowledge from different areas of mathematical analysis to analyze and solve complex problems using the principles of functional analysis,
- be able to independently identify, formulate and solve relevant problems, as well as to plan and execute qualified tasks within a given time frame.

*Judgement and Approach*

For a passing grade the doctoral student must

- be able to argue for the important role of the principles of functional analysis in different areas of research in mathematics and physics,
- be able to identify their own need for further knowledge and take responsibility for developing their own knowledge.

The course treats applications of - the Hahn-Banach theorem, weak convergence and compactness, - the Riesz representation theorem, - the use of orthonormal bases, - boundedness, compactness and spectra of integral operators, - the spectral theorem for compact, self-adjoint operators.

Lax, Peter D.: Functional Analysis. John Wiley & Sons, 2002. ISBN 9780471556046.

**Types of instruction:** Lectures, seminars

**Examination format:** Miscellaneous.
The examination consists of oral presentations of solutions of problems or proofs of relevant results during the course and a problem-solving project at the end of the course.**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** The postgraduate student is assumed to have knowledge corresponding to MATP35F Linear Functional Analysis or be studying that course in parallel with MATP45F.

**Course coordinators:** **Web page:** http://www.ctr.maths.lu.se/course/MATP45/