Functional Analysis

Course

URL study guide

https://studiegids.vu.nl/en/courses/2024-2025/XB_0079

Course Objective

At the end of this course the students will be able to… …describe the basic theory of (infinite dimensional) complete normed vector spaces, so-called Banach spaces; …apply fundamental results about bounded linear operators between Banach spaces with special emphasis on bounded linear functionals: the Hahn-Banach extension theorem, the uniform boundedness principle, the open mapping and the closed graph theorem, compactness of the unit ball with respect to weak topologies; …use orthonormal bases and projection operators as additional tools to investigate the class of Hilbert spaces, namely those Banach spaces whose norm is induced by an inner product; …generalize the notion of eigenvalues and eigenvectors for bounded linear operators on (infinite dimensional) Hilbert spaces and describe more precisely the spectral properties of compact self-adjoint operators; …recognize the above abstract framework at play in several interesting concrete examples: Lp spaces, approximation of functions by polynomials and Fourier series, solutions of integral and differential equations; …solve exercises both on the general theory of Banach and Hilbert spaces and on its application to concrete function spaces.

Course Content

Infinite dimensional vector spaces such as the space of continuous or differentiable functions on an interval play a crucial role in analysis and in the theory of integral and differential equations. These spaces are often endowed with a natural norm (think of the uniform norm of a continuous function) which enables us to speak of convergence of sequences and leads us to the crucial question about how to best approximate a function by simpler ones like polynomials or Fourier sums. In this course we will put the above problems in the general framework of Banach spaces and develop a general theory for these spaces and their linear transformations. A special place in our discussion will be occupied by Hilbert spaces, where the norm comes from an inner product. In this situation, we will have additional tools to approximate elements and to study the eigenvalues and eigenvectors of linear operators. More specifically, we will learn the following topics during the course: 1) Basic theory and examples of Banach spaces 2) The space of bounded linear operators between Banach spaces 3) The dual of a Banach space 4) Extension of linear functionals: The Hahn-Banach theorem 5) The uniform boundedness principle 6) The open mapping and the closed graph theorem 7) Weak and weak
- topology 8) Reflexive and separable Banach spaces 9) Basic theory of Hilbert spaces: orthonormal basis and projection operators on convex sets 10) Linear bounded operators on Hilbert spaces: adjoint and spectrum 11) The spectral theorem for self-adjoint compact operators 12) Applications to the solution of integral and differential equations

Teaching Methods

Lectures and exercise sessions, both 2 hours per week.

Method of Assessment

For this course there are 5 hand-in assignments (total: 15%), a midterm exam (35%) and a final exam (50%). There will also be a resit examination: the final grade will then be determined either (i) by the grade of the resit (85%) and the grade of the hand-in assignments (15%), or (ii) by the grade of the resit alone (100%), depending which of the two options result in a higher grade.

Literature

Mandatory literature: - A concise introduction to the subject is Bryan P. Rynne and Martin A. Youngson, ‘Linear Functional Analysis’, 2nd Edition, Springer, 2008, ISBN 978-1-84800-005-6. - A book with many motivating examples covering all material from scratch in detail is Erwin Kreyszig, ‘Introductory Functional Analysis with Applications’, John Wiley & Sons, 1978, ISBN 0-471-50731-8.

Target Audience

Bachelor Mathematics Year 3

Additional Information

This course offers an honours extension. Please have a look at the dedicated page for more information.

Recommended background knowledge

The following background is mandatory: Single variable calculus, Linear algebra, and Mathematical analysis. It is recommended to revise the basic notions of metric spaces before the course starts. These can be found in the first chapter of either the two books in the literature. Examples from measure theory will not be studied in this course. Therefore, no prior knowledge of measure theory is needed.
Academic year1/09/2431/08/25
Course level6.00 EC

Language of Tuition

  • English

Study type

  • Premaster
  • Bachelor