### Functional Analysis - 8EC

The course starts with a thorough introduction to topological vector spaces (TVS). In general TVS, a norm — and even a metric — is not available. However, many theorems well-known in the Banach space context still hold. Their proofs then take a much more topological flavour.

To smoothen the transition to non-metric vector spaces, we start by proving the Banach-Steinhaus Theorem (a.k.a. Uniform Boundedness Principle), first in the Banach space setting, then in the more general TVS context. This result is based on the Baire Category Theorem, other important consequences of which are also presented, such as the Open Mapping Theorem and the Closed Graph Theorem. We then study in more detail the important case of locally convex TVS, which are most common in applications, e.g. for the theory of distributions, which is briefly discussed.

The next topic is convexity and various versions of the Hahn-Banach Theorem. This is followed by the introduction of weak and weak* topologies, and related compactness results, such as the Banach-Alaoglu Theorem and the Eberlein-Smulyan Theorem.

In the following part of the course, we study compact operators on Banach spaces. We introduce the notions of spectrum and eigenvalues in this setting, and prove the Spectral Theorem for compact operators. We discuss the approximation property of Banach spaces.

With the aim of generalizing the Spectral Theorem to other classes of operators, we then have a look at Banach algebras and explain what the spectrum in this context means. We introduce a symbolic calculus on Banach algebras by means of Cauchy's Integral Theorem.

The topic of the next lectures is Gelfand theory on commutative Banach algebras. This includes the introduction of the Gelfand transform as a generalization of the Fourier transform, and the Gelfand-Naimark Theorem. We discuss applications of the theory, such as a Spectral Theorem and a functional calculus for bounded normal operators on Hilbert spaces.

The final part of the course deals with unbounded operators in Hilbert spaces, and in particular the notions of symmetric and selfadjoint operators. Selfadjointness for unbounded operators is a delicate matter, related to the domain of the operator. The theory of selfadjoint extensions of symmetric operators is presented, with concrete examples of differential operators discussed in the homework problems.

We then study the Spectral Theorem for unbounded selfadjoint operators. The proof we present, due to John von Neumann, is based on the Cayley transform. A notion of integration with respect to a spectral family is central to the Spectral Theorem. In the case of unbounded operators, it is deeply connected with the Lebesgue-Stieltjes integral. As an application, we conclude the course with Stone's theorem, which characterizes one-parameter groups of unitary operators. If time permits, we also briefly discuss applications to quantum mechanics.

Prerequisites

Basic knowledge of bounded linear operators in Banach and Hilbert spaces, of general topology and metric spaces. Students should be familiar with notions such as Cauchy sequences in normed vector spaces, operator norm, dual space, inner product and Cauchy-Schwarz inequality, orthogonal projections in Hilbert spaces, orthonormal basis and Fourier coefficients, adjoint operator, selfadjoint, unitary and normal operators. This course is not suitable as a first acquaintance with functional analysis. 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.