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Preface p. xi Introduction p. xiii Part I Hilbert Spaces and Basic Operator Theory p. 1 1 Linear spaces; normed spaces; first examples p. 3 1.1 Linear spaces p. 3 1.2 Normed spaces; first examples p. 7 1.2a Holder's inequality p. 9 1.2b Minkowski's inequality p. 11 1.3 Topological and geometric notions p. 12 1.4 Quotient normed space p. 14 1.5 Completeness; completion p. 16 1.6 Exercises p. 21 2 Hilbert spaces p. 25 2.1 Basic notions; first examples p. 25 2.1a Cauchy-Schwartz inequality and the Hilbertian norm p. 26 2.1b Bessel's inequality p. 29 2.1c Complete systems p. 29 2.1d Gram-Schmidt orthogonalization procedure; orthogonal bases p. 30 2.1e Parseval's identity p. 32 2.2 Projection; orthogonal decompositions p. 34 2.2a Separable case p. 34 2.2b The distance from a point to a convex set p. 35 2.2c Orthogonal decomposition p. 35 2.3 Linear functionals p. 37 2.3a Linear functionals in a general linear space p. 37 2.3b Bounded linear functionals p. 38 2.3c Bounded linear functionals in a Hilbert space p. 40 2.3d An example of a non-separable Hilbert space p. 41 2.4 Exercises p. 41 3 The dual space p. 47 3.1 The Hahn-Banach theorem and its first consequences p. 47 3.1a Corollaries of the Hahn-Banach theorem p. 48 3.2 Examples of dual spaces p. 51 3.3 Exercises p. 52 4 Bounded linear operators p. 55 4.1 Completeness of the space of bounded linear operators p. 55 4.2 Examples of linear operators p. 57 4.3 Compact operators p. 59 4.3a Compact sets p. 59 4.3b The space of compact operators p. 62 4.4 Dual operators p. 63 4.5 Operators of finite rank p. 65 4.5a Compactness of the integral operator in L[subscript 2] p. 66 4.6 Convergence in the space of bounded operators p. 67 4.7 Invertible operators p. 68 4.8 Exercises p. 69 5 Spectrum. Fredholm theory of compact operators p. 75 5.1 Classification of spectrum p. 75 5.2 Fredholm theory of compact operators p. 77 5.3 Exercises p. 83 6 Self-adjoint operators p. 87 6.1 General properties p. 87 6.2 Self-adjoint compact operators p. 89 6.2a Spectral theory p. 89 6.2b Minimax principle p. 92 6.2c Applications to integral operators p. 94 6.3 Order in the space of self-adjoint operators p. 96 6.3a Properties of the ordering p. 96 6.4 Projection operators p. 100 6.4a Properties of projections in linear spaces p. 100 6.4b Orthoprojections p. 101 6.5 Exercises p. 103 7 Functions of operators; spectral decomposition p. 105 7.1 Spectral decomposition p. 108 7.1a The main inequality p. 109 7.1b Construction of the spectral integral p. 110 7.2 Hilbert theorem p. 111 7.3 Spectral family and spectrum of self-adjoint operators p. 114 7.4 Simple spectrum p. 116 7.5 Exercises p. 117 Part II Basics of Functional Analysis p. 119 8 Spectral theory of unitary operators p. 121 8.1 Spectral properties of unitary operators p. 121 8.2 Exercises p. 126 9 The fundamental theorems and the basic methods p. 127 9.1 Auxiliary results p. 128 9.2 The Banach open mapping theorem p. 131 9.3 The closed graph theorem p. 132 9.4 The Banach-Steinhaus Theorem p. 135 9.5 Bases in Banach spaces p. 139 9.6 Linear functionals; the Hahn-Banach theorem p. 142 9.7 Separation of convex sets p. 146 9.8 The Eberlain-Schmulian theorem p. 152 9.9 Extremal points; the Krein-Milman theorem p. 153 9.10 Exercises p. 159 10 Banach algebras p. 167 10.1 Preliminaries p. 167 10.2 Gelfand's theorem on maximal ideals p. 171 10.3 Analytic functions p. 172 10.4 Gelfand map; the space of maximal ideals p. 176 10.4a The space of maximal ideals p. 179 10.5 Radicals p. 181 10.6 Involutions; the Gelfand-Naimark theorem p. 184 10.7 Application to spectral theory p. 189 10.8 Application to a generalized limit and combinatorics p. 193 10.9 Exercises p. 196 11 Unbounded self-adjoint and symmetric operators in H p. 203 11.1 Basic notions and examples p. 203 11.2 More properties of symmetric operators p. 210 11.3 The spectrum [gs](A) p. 211 11.4 Elements of the "graph method" p. 215 11.5 Cayley transform; spectral decomposition p. 216 11.6 Symmetric and self-adjoint extensions of a symmetric operator p. 221 11.7 Exercises p. 225 A Solutions to exercises p. 227 A.1 Solutions to the exercises of Chapter 1 p. 227 A.2 Solutions to the exercises of Chapter 2 p. 235 A.3 Solutions to the exercises of Chapter 3 p. 250 A.4 Solutions to the exercises of Chapter 4 p. 254 A.5 Solutions to the exercises of Chapter 5 p. 263 A.6 Solutions to the exercises of Chapter 6 p. 270 A.7 Solutions to the exercises of Chapter 7 p. 277 A.8 Solutions to the exercises of Chapter 8 p. 279 A.9 Solutions to the exercises of Chapter 9 p. 282 A.10 Solutions to the exercises of Chapter 10 p. 296 A.11 Solutions to the exercises of Chapter 11 p. 309 Bibliography p. 311 Symbols index p. 313 Subject index