![]() $T^*$ is always closed in the weak $^*$-topology. $T^*$ is the graph of an operator when $(0,y^*)\in G(T^*)\implies y^*=0$, equivalently, when $T$ is densely defined. Identify $(E_1\oplus E_2)^*$ with $E_1^*\oplus E_2^*$ so the annihilator of $G(T)$ is In terms of the graph of the operators, this means that $(x^*,y^*)\in G(T^*)$ exactly when ![]() If $T: E_1 \supseteq D(T)\rightarrow E_2$ is a linear map between Banach spaces, then we define $x^*\in D(T^*)$ with $T^*(x^*)=y^*$ to mean that $y^*(x) = x^*(T(x))$ for each $x\in D(T)$. 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.ĭr.You can use essentially the same definition. A discussion of spectral measures, the Borel functional calculus, and (a version of) the spectral theorem concludes the part on C*-algebras.įinally, a brief introduction to the theory of unbounded operators on Hilbert spaces will be given. 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. As a consequence we get the continuous functional calculus for normal elements in arbitrary 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). ![]() 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.Īfter a few general results on C*-algebras we proceed with commutative C*-algebras. We briefly discuss the Approximation Property. 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. Next we study compact operators on Banach spaces. The weak and weak* topologies are discussed, as are the Banach-Alaoglu theorem, the Eberlein-Smulian theorem, and the Krein-Milman theorem. For locally convex spaces we consider the Hahn-Banach theorem and other separation results. Important consequences of this theorem are the Open Mapping Theorem, Closed Graph Theorem, Bounded Inverse Theorem, and Uniform Boundedness Principle. We will discuss metrizability and completeness and the Baire Category Theorem. Locally convex spaces, which are topological vector spaces where the topology is generated by a collection of seminorms, will get special attention. The course starts with a thorough introduction to topological vector spaces. 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. 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. 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. ![]() It is essential that students who haven't yet had a course in measure and integration theory follow one in parallel to our course. Later on, we will however assume that all participants are familiar with measure and integration theory at a workable level. Measure and integration theory is not a formal prerequisite, an intuitive knowledge of it will be enough in the beginning of the course. Knowledge of compact operators or reflexivity (topics covered in some introductory courses) is not a prerequisite. 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. 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. 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. ![]()
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