**Functional Analysis [Fall 2019]**

**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, we will however assume that all participants are familiar with measure and integration theory at a workable level. It is essential that students who haven't yet had a course in measure and integration theory follow one in parallel to our 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 technical proofs, many of which will be sketched or omitted. The main topics are topological vector spaces, compact operators, Banach algebras, commutative and non-commutative C*-algebras and their representations and spectral theory, and unbounded operators.

The course starts with a thorough introduction to topological vector spaces. Locally convex spaces, which are topological vector spaces where the topology is generated by a collection of seminorms, will get special attention. We will discuss metrizability and completeness and the Baire Category Theorem. Important consequences of this theorem are the Open Mapping Theorem, Closed Graph Theorem, Bounded Inverse Theorem, and Uniform Boundedness Principle. For locally convex spaces we consider the Hahn-Banach theorem and other separation results. The weak and weak* topologies are discussed, as are the Banach-Alaoglu theorem, the Eberlein-Smulian theorem, and the Krein-Milman theorem.

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.

After a few general results on C*-algebras we proceed with commutative C*-algebras. We introduce the Gelfand transform and discuss the commutative Gelfand Naimark Theorem on representation of a unital commutative C*-algebra as a C(X). As a consequence we get the continuous functional calculus for normal elements in arbitrary C*-algebras. The famous Gelfand-Naimark theorem which states that the closed *-invariant subalgebras of B(H) are the only C*-algebras, up to isometric isomorphism, is treated next. A discussion of spectral measures, the Borel functional calculus, and (a version of) the spectral theorem concludes the part on C*-algebras.

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, approximate point spectrum, Fredholm operators and Fredholm index, 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

- Docent: Marcel de Jeu