Entry requirements

120 credits including 90 credits in mathematics with Real Analysis.

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;
• 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.