Prerequisites
Basic knowledge of metric spaces and general topology (and preparedness to fill gaps). This course is not suitable as a first encounter with functional analysis! It is expected that you have successfully taken a Bachelor course on functional analysis and have basic knowledge of Banach and Hilbert spaces and bounded linear operators between them, including the core results. The book by Rynne and Youngson used at several Dutch universities gives some indication.
Here are some keywords to test yourself: Cauchy sequence, norms and their equivalence, operator norm, dual space, transpose (or adjoint) of an operator between Banach spaces, Hahn-Banach theorem. Baire's (category) theorem and its consequences (open mapping and closed graph theorems, uniform boundedness principle). Weak and weak-* topologies and Alaoglu's theorem. Inner products, Cauchy-Schwarz inequality, orthogonal decomposition of a Hilbert space related to a closed subspace, orthonormal basis, orthogonal projection, adjoint of an operator, self-adjoint/unitary/normal operators. In depth knowledge of spectral theory, compact operators or reflexivity (topics covered in some introductory courses) is very helpful, but not a prerequisite.
Measure and integration theory is not a formal prerequisite; an intuitive knowledge of it will be enough in the beginning of the course. Later on, however, it will be assumee that all participants are familiar with measure and integration theory at a workable level. It is strongly recommended that students who have not yet had a course in measure and integration theory follow one in parallel to this course. Complex analysis has extensive connections with functional analysis, but they will not be very essential for us.
Aim of the course
This course provides a broad basis in functional analysis well beyond the introductory level, preparing for a specialization in
analysis as well as developing the tools for advanced applications of functional analysis in other disciplines. The main topics are topological vector spaces, compact operators, Banach and C*-algebras, characters and (a bit) on representations, and commutative C*-algebras and spectral theory.
The course starts with a very rapid summary of the assumed background in order to accommodate diverse backgrounds and fill in some topics you might not have seen. We plan to briefly touch upon a few topics in Banach space theory that tend to be neglected in the non-specialized literature, like unconditional convergence of series, (un)complemented subspaces and the Approximation Property. We will then give an introduction to topological vector spaces with special emphasis on locally convex spaces. These are topological vector spaces whose topology is generated by a collection of seminars. We consider the extensions of the classical theorems, i.e. the Hahn-Banach (separation) theorem, open mapping theorem, Alaoglu, to their natural habitats (which is locally convex spaces for Hahn-Banach, but not for the other results). The weak and weak-* topologies are discussed, as are the Banach-Alaoglu theorem, the Eberlein-Smulian theorem, the Krein-Milman theorem, and reflexivity.
Next we study compact operators on Banach spaces. We show that they constitute a two-sided ideal in the bounded operators, that
compactness of an operator is equivalent to compactness of its adjoint, and we present the Riesz-Schauder theory on the spectrum of a
compact operator. In this context we also touch upon Fredholm operators and the Fredholm index.
We introduce the notion of spectrum of an element in an arbitrary Banach algebra. Specializing to the complex case, we prove nonempty-ness and the spectral radius formula. We briefly look at the Riesz (holomorphic) functional calculus and the spectral mapping theorem. (We also define the notions of discrete, continuous and residual spectrum for bounded operators and the extent to which they can be adapted to Banach algebras.) We then study the Gelfand transform of a commutative Banach algebra.
We then move on to C*-algebras. After preliminary results on general C*-algebras, we cover Gelfand's theorem that asserts that the Gelfand transform is an isometric *-isomorphism between a unital commutative C*-algebra and the continuous functions on its maximal ideal space. As an application to non-commutative C*-algebras we obtain the continuous functional calculus for normal elements of arbitrary C*-algebras. Time permitting, we may turn to the Gelfand-Naimark theorem to the effect that every C*-algebra is isometrically *-isomorphic to a closed *-invariant subalgebra of B(H) for some Hilbert space H and/or to the spectral theorem for normal operators on Hilbert spaces.
Finally, a brief introduction to the theory of unbounded operators on Hilbert spaces will be given.
Homework
There will be 7 homework sets, assigned every two weeks. They will be more substantial than weekly sets would be, but not twice as much. Expect 5 or 6 exercises per set. Your homework mark will be based on your best 6 sets.
Examination
There will be a written exam lasting 3 hours (or a bit more if you are eligible). No electronic devices of any sort will be
allowed. You can bring Conway's book and my lecture notes. The same rules will apply to the retake. (The latter might be oral if there are very few participants.) In order to pass the course you must achieve at least a 5.0 for the exam (or the retake). If this is the case and the mark for the homework mark is higher than that for the exam, the final mark is calculated as the weighted average, where the exam contributes 80% and the homework 20%. Otherwise, the mark for the exam will be the final mark.
Literature
Required: John B. Conway: A course in functional analysis. 2nd edition, Springer, 2007. [A very widely used
middle-of-the-road approach.]
I strongly recommend to look also at a few other sources for different perspectives! For example:
Michael Müger: Introduction to functional analysis. Lecture notes used for several Bachelor courses with much
material beyond Bachelor level. Available at https://www.math.ru.nl/~mueger/FA-notes.pdf
Gert K. Pedersen: Analysis NOW. Corrected 2nd printing, Springer, 1995. [A crisp no-nonsense approach by
a prominent operator algebraist. (Not much on locally convex spaces.) The first chapter is a very useful
summary of the relevant general topology.]
Walter Rudin: Functional analysis. 2nd edition, McGraw-Hill, 1991. [This book takes the top-down approach,
starting from general topological vector spaces. It covers distributions in quite some detail, a topic in locally
convex spaces that is very important in applications.]
Peter D. Lax: Functional analysis. Wiley, 2002. [Beautiful account by a prominent analyst for whom there was
no dividing line between pure and applied mathematics.]
Lecturer
Michael Müger (Radboud University), email: michael.mueger@ru.nl
Teaching assistants
Jort Jacobs, email: jort.jacobs@ru.nl
Aljoscha Melssen, email: aljoscha.melssen@ru.nl
- Docent: Michael Mueger