Course Description:

The subject of Functional Analysis originated in the study of integral/differential equations and, more generally, equations involving operators on infinite dimensional spaces. These techniques can be helpful, for instance, in analysing convergence series and can be used to make sense of "eigenvalues" and "determinants" for infinite-dimensional matrices. Functional Analysis has found broad applicability in diverse areas of Mathematics, Physics, Economics, and other sciences. Students will be introduced to the theory of Banach and Hilbert spaces. We will discuss such fundamental theorems of the Functional Analysis as Hahn-Banach Theorem, Uniform Boundedness Theorem, Open Mapping Theorem, Closed Graph Theorem, etc. and develop the Spectral Theory of linear operators.

Exam: written, closed book.

Homework:

Weekly homeworks will be assigned,  and it essential to solve these problem to gain understanding and prepare for the exam. However, the solutions will not be corrected. The solutions will be discussed at the exercise classes. 

To pass the course, each student muss present solutions of at least two homework problems from two different exercise sheets during the exercise classes.

References:

• J. Conway, A Course in Functional Analysis, Springer, 1994.
• D. Lax, Functional Analysis, Wiley Interscience, New York, 2002.
• M. Reed and B. Simon, Methods of modern mathematical physics, Volume 1: Functional analysis, Academic Press, 1981.
• W. Rudin, Functional Analysis, McGraw-Hill, 1991.
• R. Zimmer, Essential Results in Functional Analysis, University of Chicago Press, 1990.