Functional Analysis, 7.5 Credits

Contents

The course covers basic functional analysis. The theory includes linear operators on Hilbert spaces, Banach spaces, and locally convex spaces. It also includes an introduction to spectral theory and fixed point theorems, and applications of  these on non-linear operators. Function spaces and base expansions of different kinds are also discussed.

Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• define and thoroughly explain Banach and Hilbert spaces and self-adjoint operators
• independently prove and thoroughly explain central theorems

    Skills

• apply the spectral theorem for compact self-adjoint operators and Fredholm’s alternative, and decide which properties an operator has
• apply Hilbert space-theory, including Riesz’ representation theorem and weak convergence, and critically reflect over chosen strategies and methods in problem solving
• independently decide if a linear space is a Banach space
• identify and independently use contractions of Banach spaces via Brouwers and Schauders fixed point theorems

  Judgement and approach

• from a critical point of view analyse operators from applications

Required Knowledge

The course requires a course in Real Analysis on advanced level. Proficiency in English equivalent to Swedish upper secondary course English 5/A. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

Form of instruction

Teaching is mainly in the form of lectures.

Examination modes

The course is examined by a written exam. For the course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG).

A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade. A student who has not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, chapter 6, 22§). A request to be assigned another examiner should be addressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for at least two years following the student's first participation in the course.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

Other regulations

This course can not be included in a degree together with another course with similar contents. When in doubt, the student should consult the director of study at the department of mathematics and mathematical statistics.