MAT4400 – Linear Analysis with Applications

Schedule, syllabus and examination date

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Exams after the reopening

As a general rule, exams will be conducted without physical attendance in the autumn of 2021, even after the reopening. See the semester page for information about the form of examination in your course. See also more information about examination at the MN Faculty in 2021.

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.

Admission to the course

Students admitted at UiO must apply for courses in Studentweb. Students enrolled in other Master's Degree Programmes can, on application, be admitted to the course if this is cleared by their own study programme.

Nordic citizens and applicants residing in the Nordic countries may apply to take this course as a single course student.

If you are not already enrolled as a student at UiO, please see our information about admission requirements and procedures for international applicants.

Overlapping courses


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.


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. The assignment must be written with a typesetting software for mathematics, e.g. LaTeX.

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: MAT3400 – 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.

Grading scale

Grades are awarded on a scale from A to F, where A is the best grade and F is a fail. Read more about the grading system.

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) Oct. 20, 2021 2:32:47 PM

Facts about this course

Teaching language