Linear Analysis, 7.5 credits

Learning outcomes

Knowledge and understanding
After completed studies, the student shall

• be able to explain basic concepts in linear functional analysis and how they are related, and
• be able to formulate and prove basic theorems in linear functional analysis.

Competence and skills
After completed studies, the student shall

• be able to apply definitions, theorems and methods in linear functional analysis in order to solve problems in linear functional analysis, and
• be able to present well-structured solutions for problems in functional analysis.

Content

Metric spaces. Convergence and completeness. Linear and bounded maps on Banach spaces and Hilbert spaces. Banach fixed point theorem and applications. Linear functionals and Reisz' representation theorem. Orthogonal projection and othonormal systems in Hilbert spaces. Spectral theorem for compact and self-adjoint operators on Hilbert spaces. Unbounded operators. The Fourier transform. Applications on integral and differential equations.

Examinations and grades

Examination, 7.5 credits (Code: A002)
Grades used are Fail (F), Sufficient (E), Satisfactory (D), Good (C), Very Good (B) or Excellent (A).

According to the Higher Education Ordinance, Chapter 6, Section 18, a grade is to be awarded on the completion of a course, unless otherwise prescribed by the university. The university may determine which grading system is to be used. The grade must be determined by a teacher specifically nominated by the university (the examiner).

In accordance with university regulations on grading systems for first and second-cycle courses and study programmes (Vice-Chancellor’s decision ORU 2018/00929), one of the following grades is to be used: fail (U), pass (G) or pass with distinction (VG). For courses included in an international master’s programme (60 or 120 credits) or offered to the university’s incoming exchange students, the A to F grading scale is to be used. The vice-chancellor, or a person appointed by them, may decide on exceptions from this provision for a specific course, if there are special grounds for doing so.

The grades used on this course are Fail (F), Sufficient (E), Satisfactory (D), Good (C), Very Good (B) or Excellent (A).

Comments on grades

Grading scale A-F according to the vice-rector's decision 2019-11-12 case no: ORU 06367/2019.

Modes of assessment

• Examination (code A002): Written examination

For students with a documented disability, the university may approve applications for adapted or other modes of assessment.

For further information, see the university's local examination regulations.

Specific entry requirements

Linear Algebra II, 7.5 Credits, Complex Analysis, 6 Credits, Differential Equations, 7.5 Credits.

For further information, see the university's admission regulations.

Other provisions

All or part of the course may be given in English.

Students who have been admitted to and registered on a course have the right to receive tuition and/or supervision for the duration of the time period specified for the particular course to which they were accepted (see, the university's admission regulations (in Swedish)). After that, the right to receive tuition and/or supervision expires.

Reading list and other learning resources

Required Reading

Debnath, Lokenath and Mikusinski, Piotr (2005)
Hilbert Spaces with Applications
Elsevier Academic Press

Additional materials are provided by the Mathematics Unit.