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.
Grading scale: Failed, pass
Examiner:

Admission Details

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 Occasion Information

Contact and Other Information

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