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Functional Analysis Page 1 List of contents 1.    Hilbert spaces   31.1 Basic definitions and results   3 1.2 Orthogonality and orthonormal bases   81.3 Isomorphisms 16

2.    Bounded linear operators 18 2.1 Bounded linear mappings 182.2 Adjoint operators 25

2.3 Projection operators 302.4 Baire's Category Theorem and Banach-Steinhaus-Theorem 34

3. Spectral analysis of bounded linear operators 363.1 The order relation for bounded selfadjoint operators 36 3.2 Compact operators on a Hilbert space 433.3 Eigenvalues of compact operators 51 3.4 The spectral decomposition of a compact linear operator 58

Functional Analysis Page 2 Introduction to Spectral Theory in Hilbert Space The aim of this course is to give a very modest introduction to the extremely rich and well-developed theory of Hilbert spaces, an introduction that hopefully will provide the students with a knowledge of some of the fundamental results of the theory and will make themfamiliar with everything needed in order to understand, believe and apply the spectral theorem for selfadjoint operators in Hilbert space. This implies that the course will have to giveanswers to such questions as -  What is a Hilbert space? -  What is a bounded operator in Hilbert space? -  What is a selfadjoint operator in Hilbert space? -  What is the spectrum of such an operator? -  What is meant by a spectral decomposition of such an operator?

LITERATURE: -  English:* 

G. Helmberg: Introduction to Spectral Theory in Hilbert space(North-Holland Publishing Comp., Amsterdam-London)

*  R. Larsen: Functional Analysis, an introduction(Marcel Dekker Inc., New York)

*  M. Reed and B. Simon: Methods of Modern Mathematical Physics I:Functional Analysis

(Academic Press, New York-London) -  German:* 

H. Heuser: Funktionalanalysis, Theorie und Anwendung(B.G. Teubner-Verlag, Stuttgart)

Functional Analysis Page 3 Chapter 1: Hilbert spaces Finite dimensional linear spaces (=vector spaces) are usually studied in a course calledLinear Algebra or Analytic Geometry, some geometric properties of these spaces may also have been studied, properties which follow from the notion of an angle being implicit in thedefinition of an inner product. We shall begin with some basic facts about Hilbert spaces including such results as the Cauchy-Schwarz inequality and the parallelogram andpolarization identity

$1 Basic definitions and results (1.1)  Definition: A linear space E over K c {R,C} is called an inner product space (or apre-Hilbert space) over K that if there is a mapping ( 

x ): E%E t K that satisfies thefollowing conditions:

(S1)   (xxx)m0 and (xxx)=0 if and only if x=0(S2)   (x+y