### Functional Analysis - M1 - 8EC

[FALL 2020]

Prerequisites

Basic knowledge of Banach and Hilbert spaces and bounded linear operators as is provided by introductory courses, and hence also of general topology and metric spaces. Keywords to test yourself: Cauchy sequence, equivalence of norms, operator norm, dual space, adjoint operator, Hahn-Banach theorems, Baire category theorem, closed graph theorem, open mapping theorem, uniform boundedness principle, inner product and Cauchy-Schwarz inequality, orthogonal decomposition of a Hilbert space related to a closed subspace, orthonormal basis and Fourier coefficients, adjoint operator, orthogonal projection, selfadjoint/unitary/normal operators. You should be familiar with these notions and results at a workable level before you take this course, which is not suitable as a first acquaintance with functional analysis. Knowledge of compact operators or reflexivity (topics covered in some introductory courses) is 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, we will assume 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.

Aim of the course

This course provides a broad basis in functional analysis well beyond the introductory level, preparing for a specialization in fundamental analysis as well as developing the tools for advanced functional analytic applications in other disciplines. In order to cover a wide range of diverse topics, the lectures will often focus more on conceptual aspects rather than on technical proofs, many of which can only be sketched or even have to be omitted. The main topics are topological vector spaces, compact operators, Banach algebras, C*-algebras and their representations, and commutative C*-algebras and spectral theory.

The course starts with an introduction to topological vector spaces. Locally convex spaces, which are topological vector spaces where the topology is generated by a collection of seminorms, will receive considerable attention. For locally convex spaces, we consider Hahn-Banach-type theorem and separation 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. We briefly discuss the Approximation Property.

The notion of spectrum is introduced for an element in an arbitrary Banach algebra and we also present the Riesz functional calculus and the spectral mapping theorem. There will also be a discussion of the approximate point spectrum, Fredholm operators and the Fredholm index.

We then move on to C*-algebras. After preliminary results on general C*-algebras, we cover the commutative Gelfand-Naimark theorem. It 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 a consequence, we obtain the continuous functional calculus for normal elements of arbitrary C*-algebras. The famous Gelfand-Naimark theorem is treated next. It states that every C*-algebra is isometrically *-isomorphic to a closed *-invariant subalgebra of B(H) for some Hilbert space H. We conclude the part on C*-algebras with spectral measures associated with representations of commutative C*-algebras, and with the ensuing Borel functional calculus. The spectral theorem for normal operators then results immediately, as a special case of a more general theory.

Finally, a brief introduction to the theory of unbounded operators on Hilbert spaces will be given.

If time permits, also topics such as ergodic theory,  Sturm-Liouville operators and the Fourier transform for general locally compact abelian groups will be addressed.

Lecturers

dr. Marcel de Jeu (Leiden University), mdejeu@math.leidenuniv.nl
dr.ir. Onno van Gaans (Leiden University), vangaans@math.leidenuniv.nl