Functional Analysis and Harmonic Analysis
Funktionalanalys och harmonisk analys

FMAN80, 7.5 credits, A (Second Cycle)

Valid for: 2024/25
Faculty: Faculty of Engineering LTH
Decided by: PLED F/Pi
Date of Decision: 2024-04-15
Effective: 2024-05-08

General Information

Depth of study relative to the degree requirements: Second cycle, in-depth level of the course cannot be classified
Elective for: F5, F5-tf, F5-bs, F5-ss, MMSR2, Pi4-bs, Pi4-ssr
Language of instruction: The course will be given in English on demand

Aim

Functional analysis and harmonic analysis are fundamental tools in many mathematical applications, e.g., in field theory, solid mechanics, control theory, signal processing and in mathematical statistics and numerical analysis. The aim of the course is to convey knowledge about basic concepts and methods in the subject, and to give the ability, both to follow discussions where these are used and to independently solve mathematical problems 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 apparently different fields.

Learning outcomes

Knowledge and understanding
For a passing grade the student must

• have knowledge of different common norms and their usefulness in different applications.
• be familiar with the meaning of the concepts of completeness and compactness, and be aware that the choice of norm has an impact on these properties of a space. In particular, the student should have a good understanding of the difference between the infinite and the finite dimensional case.
• be familiar with the basic properties of the dual space of a normed space.
• be familiar with the most common classes of linear and bounded operators, and have an understanding of how spectral theory provides information on the properties of a linear operator.
• be familiar with how the properties of a function are reflected in its Fourier transform.
• be able to explain the basic theory in an oral examination.

Competences and skills
For a passing grade the student must

• with access to literature independenly be able to integrate methods and approaches from different parts of the course in order to solve problems and answer questions within the framework of the course.
• in writing and orally, with proper terminology and clear logic be able to explain the solution to a mathematical problem within the course.
• be able to translate concrete mathematical problems into the abstract language of the course.

Judgement and approach
For a passing grade the student must

Contents

Functional analysis: norms and approximation,  completeness, compactness, function spaces, Hilbert spaces, orthogonality and orthogonal systems. Linear operators, compact operators and self-adjoint operators. Spectral theory. Dual spaces and Hahn-Banach's theorem.

Harmonic analysis: The Fourier transform. Sobolev spaces and the Sobolev embedding theorem.

Examination details

Grading scale: TH - (U, 3, 4, 5) - (Fail, Three, Four, Five)
Assessment: Take-home exam followed by an oral exam.

The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.

Modules
Code: 0117. Name: Functional Analysis and Harmonic Analysis.
Credits: 7.5. Grading scale: TH - (U, 3, 4, 5).

Admission

Assumed prior knowledge: FMAN55 Applied Mathematics and FMAN70/FMAN71 Matrix Theory.
The number of participants is limited to: No
Kursen överlappar följande kurser: FMA260

Reading list

• Renardy & Rogers: An Introduction to Partial Differential Equations. Springer, 2004, ISBN: 0-387-00444-0.
• Material produced at the department.
• Kreyszig, Erwin: Introductory functional analysis with applications. New York : Wiley, 1978, ISBN: 0471507318.

Contact

Course coordinator: Studierektor Anders Holst, Studierektor@math.lth.se
Teacher: Jacob Stordal Christiansen, jacob_stordal.christiansen@math.lth.se
Course homepage: https://canvas.education.lu.se/courses/20321