MAT4400 – Linear analysis with applications
Schedule, syllabus and examination date
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.
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.
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If you are not already enrolled as a student at UiO, please see our information about admission requirements and procedures.
Recommended previous knowledge
- 10 credits overlap with MAT3400 – Linear analysis with applications
- 5 credits overlap with MAT4300 – Measure and integration (discontinued)
- 5 credits overlap with MAT4340 – Elementary functional analysis (discontinued)
- 5 credits overlap with MAT3300 – Measure and integration (discontinued)
*The information about overlaps for discontinued courses may not be complete. If you have questions, please contact the Department.
6 hours of lectures/exercises every week throughout the semester.
1 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.
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.
Explanations and appeals
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.
The course is subject to continuous evaluation. At regular intervals we also ask students to participate in a more comprehensive evaluation.