Entry requirements

120 credits including 90 credits in mathematics with Real Analysis. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

On completion of the course, the student should be able to:

• report on fundamental properties of Hilbert space;
• define a compact operator and report on fundamental properties of the latter;
• report on fundamental properties of Banach space;
• apply Hahn-Banach theorem, open mapping theorem, closed graph theorem and uniform boundedness principle (Banach-Steinhaus theorem);
• solve problems involving weak topology;
• solve problems in fixed points;
• solve variational problems;
• formulate the sepctral theorem;
• apply fundamental properties of unbounded operators.

Content

Hilbert spaces. Operators on Hilbert spaces. Adjoint operator. Spectral theorem for compact normal operators. Banach spaces. Linear mappings. Reflexive spaces. Fundamental theorems of functional analysis: Hahn-Banach theorem. Banach- Steinhaus theorem (uniform boundedness principle). Open mapping theorem. Closed graph theorem. Locally uniformly convex spaces.Weak convergence and other convergences of weak type. . Linear operators in Banach spaces. Asymptotic centres. Fixed point theorems. Convexity and extreme points. Ekelands variational principle. Elementary minimax theory. Semigroups in Banach spaces, the Hille-Yosida Theorem.

Instruction

Lectures and problem solving sessions.

Assessment

A small project with oral presentation.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.