Course Syllabus for

# Specialised Course in Linear Functional Analysis Fördjupningskurs till lineär funktionalanalys

## MATP45F, 7.5 credits

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

## General Information

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

## Aim

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.

## Goals

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.

## Course Contents

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.

## Course Literature

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

## Instruction Details

Types of instruction: Lectures, seminars

## Examination Details

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.