Selected topics in functional analysis

mathematics

elective courses

Functional Analysis 1000-135AF

Taking this course, the student should have basic knowledge in functional analysis which includes the notions of normed space, Banach space, Hilbert space (with the most important examples), dual space, bounded operator, adjoint operator. It is advisable to have passed the course "Functional analysis".

The goal of this course is to present various examples of applications of tools and methods of functional analysis in other branches in mathematics. We will present spectral theory for compact operators on Banach spaces and normal operators on Hilbert spaces, and its importance in differential equations. We will also discuss Fourier transform, theory of distribution, convolution algebras, as well as weak and weak* topologies on topological vector spaces with some examples of their natural appearance.

1. Spectral properties of compact operators on Banach spaces; the Fredholm alternative and the Riesz-Schauder theorem; applications to integral equations.

2. Topological vector spaces; local convexity of topologies determined by families of seminorms; weak and weak* topologies; the Banach-Alaoglu theorem; extreme points and the Krein-Milman theorem.

3. Rudiments of theory of Banach algebras and C*-algebras; the Calkin algebra, essential spectrum and Fredholm operators; the Gelfand transform and the Gelfand-Naimark theorem.

4. Spectral measures and resolution of identity; the spectral theorem for normal operators on Hilbert spaces; functional calculus; positive and unitary operators - polar decomposition.

5. Convolution algebras - the Fourier transform as a Gelfand transform; introduction to theory of distributions - tempered distributions and the Fourier transform; Wiener's tauberian theorem and its application to the proof of the prime number theorem.

1. W. Arveson, A short course on spectral theory, Springer 2002.

2. J.B. Conway, A course in functional analysis, Springer-Verlag 1985.

3. M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler, Banach space theory, Springer 2011.

4. E. Kaniuth, A course in commutative Banach algebras, Springer 2009.

5. R.E. Megginson, An introduction to Banach space theory, Springer 1998.

6. W. Rudin, Real and complex analysis, McGraw-Hill Education, 1986

7. W. Rudin, Functional Analysis, McGraw-Hill, 1991

Knowledge and skills:

1. The student understands the Riesz theory of compact operators on Banach spaces and examples of its applications to integral equations.

2. The student can point out some natural occurences of (locally convex) linear topological spaces, as well as weak and weak* topologies in various mathematical structures.

3. The student is able to formulate and explain the spectral theorem for normal operators on a Hilbert space, an abstract approach with the aid of theory of C*-algebras and its important consequences such as functional calculus.

4. The student understands Fourier transform as an important tool occuring in various aspects, as a way of transforming "time scale" into "frequency scale", as the Gelfand transform of a convolution algebra, and as a transform acting on tempered distributions.

5. The student knows fundamentals of theory of distributions, knows the notion of tempered distribution. The students understands the idea of applying abstract functional analysis to Tauberian theorems and, in turn, to the proof of one of the most celebrated results in number theory, i.e. the prime number theorem.

Social competencies:

1. The student understands the importance of functional analysis as an abstract tool in various fields of mathematics.

The final grade based on number of points gained during classes and the score on the final exam.