MAT4410 – Advanced Linear Analysis
Course description
Schedule, syllabus and examination date
Course content
The course gives an introduction to more advanced topics in functional analysis and the theory of linear operators. Among the topics covered are: Further Banach space theory (including the Banach-Steinhaus theorem, the open mapping theorem, the closed graph theorem, including applications of these, and reflexivity); further Hilbert space theory (including adjoint operators, orthogonal projections, unitary operators, compact operators, the spectral theorem for self-adjoint, compact operators, Hilbert-Schmidt operators, and trace class operators); applications to Sturm-Liouville theory, and Fredholm theory.
Learning outcome
The course presents several concepts and results in linear analysis that are important for further studies in the field of operator algebras. It might also be useful for students aiming to specialize in?stochastic analysis or?partial differential equations.
After completing the course:?
- you will be used to work with bounded linear operators between infinite dimensional spaces
- you will have seen the importance of?completeness in connection with three fundamental theorems about operators?between Banach spaces
- you will be?acquainted with several important classes of bounded linear operators on Hilbert spaces
- you will have gained a good understanding of the spectral theorem for compact self-adjoint?operators and be able to?apply it to solve certain Sturm-Liouville problems
- you will know how the Fredholm alternative can be used to find solutions of certain integral equations
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.
Recommended previous knowledge
- MAT1110 – Calculus and Linear Algebra
- MAT1120 – Linear Algebra
- MAT2400 – Real Analysis
- MAT3400 – Linear Analysis with Applications/MAT4400 – Linear Analysis with Applications
- MAT3500 – Topology/MAT4500 – Topology (can be taken simultaneously)
Overlapping courses
- 5 credits overlap with MAT3300 – Measure and integration (discontinued).
- 5 credits overlap with MAT4300 – Measure and integration (discontinued).
- 3 credits overlap with MAT4350 – Functional analysis (discontinued).
- 2 credits overlap with MAT4340 – Elementary functional analysis (discontinued).
Teaching
4 hours of lectures/exercises per week throughout the semester.
Upon the attendance of three or fewer students, the lecturer may, in conjunction with the Head of Teaching, change the course to self-study with supervision.
Examination
Final written exam or final oral exam, which counts 100 % towards the final grade.
The form of examination will be announced by the lecturer by 1 October/1 March for the autumn semester and the spring semester respectively.
This course has 1 mandatory assignment that must be approved before you can participate in the final exam.
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 submit your response 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:
More about examinations at UiO
- Use of sources and citations
- Special exam arrangements due to individual needs
- Withdrawal from an exam
- Illness at exams / postponed exams
- Explanation of grades and appeals
- Resitting an exam
- Cheating/attempted cheating
You will find further guides and resources at the web page on examinations at UiO.