# MAT4400 – Linear analysis with applications

Choose semester

## Course content

MAT4400 gives an introduction to measure  and integration theory, and to operator theory (on Hilbert spaces). Covered topics include: elementary measure and integration theory, including convergence theorems, Lp-spaces and their completeness, and Carathéodory’s extension theorem. Adjoint operators, orthogonal projections, compact operators and Hilbert-Schmidt operators. The spectral theorem for compact self-adjoint operators. Applications to Sturm-Liouville theory and Fredholm theory.

## Learning outcome

After completing the course you:

• are used to work with sigma-algebras and with measures on sigma-algebras. In particular you are familiar with the most important sigma-algebras on the real line and with the Lebesgue measure on these;
• have a good understanding of measure spaces and of integrable functions, know how to compute the integral of many integrable functions and are acquainted with convergence theorems for integrals;
• know what an Lp-space is and what Hölder’s inequality says;
• are able to determine the adjoint of a bounded linear operator on a Hilbert space and decide if the operator is self-adjoint, and know well examples of self-adjoint operators, such as orthogonal projections onto closed subspaces;
• are familiar with compact operators and their most important properties, and are well aware of what is meant by the Fredholm alternative, in particular in connection with certain Integral equations;
• have a good understanding of the spectral theorem for compact self-adjoint operators and know how it can be used to solve certain Sturm-Liouville problems.

Students who are admitted to study programmes at UiO must each semester register which courses and exams they wish to sign up for in Studentweb.

## Teaching

6 hours of lectures/exercises every week throughout the semester.

## Examination

mandatory assignment written with a typesetting software for mathematics, e.g. LaTeX.

Final written examination.

### Examination support material

No examination support material is allowed.

### Language of examination

Subjects taught in English will only offer the exam paper in English.

You may write your examination paper in Norwegian, Swedish, Danish or English.

### Resit an examination

This course offers both postponed and resit of examination. Read more:

### Withdrawal from an examination

It is possible to take the exam up to 3 times. If you withdraw from the exam after the deadline or during the exam, this will be counted as an examination attempt.

### Special examination arrangements

Application form, deadline and requirements for special examination arrangements.

## Evaluation

The course is subject to continuous evaluation. At regular intervals we also ask students to participate in a more comprehensive evaluation.

10

Master

Every spring

Every spring

### Teaching language

English

The course may be taught in Norwegian if the lecturer and all students at the first lecture agree to it.

## Contact

Department of Mathematics