Entry requirements

120 credits including 90 credits of Mathematics. Real Analysis recommended.

Learning outcomes

In order to pass 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;
• report on fundamental examples of Banach algebras;
• 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 topologies. Linear operators in Banach spaces. Banach algebras and spectral theory for operators in a Banach space. Normal operators in Hilbert spaces. Survey of unbounded operators.

Instruction

Lectures and problem solving sessions.

Assessment

Written examination at the end of the course.