# MAT3400 – Linear Analysis with Applications

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## Changes in the course due to coronavirus

Autumn 2020 the exams of most courses at the MN Faculty will be conducted as digital home exams or oral exams, using the normal grading scale. The semester page for your course will be updated with any changes in the form of examination.

## Course content

This course 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.

In addition to fulfilling the Higher Education Entrance Qualification, applicants have to meet the following special admission requirements:

• Mathematics R1 (or Mathematics S1 and S2) + R2

And in addition one of these:

• Physics (1+2)
• Chemistry (1+2)
• Biology (1+2)
• Information technology (1+2)
• Geosciences (1+2)
• Technology and theories of research (1+2)

The special admission requirements may also be covered by equivalent studies from Norwegian upper secondary school or by other equivalent studies (in Norwegian).

## Teaching

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

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

## Examination

Final written exam 4 hours which counts 100 % towards the final grade.

This course has 1 mandatory assignment that must be approved before you can sit the final exam.

It will also be counted as one of the three attempts to sit the exam for this course, if you sit the exam for one of the following courses: MAT4400 – Linear Analysis with Applications

### Examination support material

No examination support material is allowed.

### Language of examination

Courses 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:

### Special examination arrangements, use of sources, explanations and appeals

See more about examinations at UiO

Last updated from FS (Common Student System) Dec. 3, 2020 9:20:44 PM

Credits
10
Level
Bachelor
Teaching
Spring
Examination
Spring
Teaching language
English

## Contact

Department of Mathematics