# Functional Analysis I: Linear Functional Analysis

Springer Science & Business Media, 06.02.1992 - 286 Seiten
Up to a certain time the attention of mathematicians was concentrated on the study of individual objects, for example, specific elementary functions or curves defined by special equations. With the creation of the method of Fourier series, which allowed mathematicians to work with 'arbitrary' functions, the individual approach was replaced by the 'class' approach, in which a particular function is considered only as an element of some 'function space'. More or less simultane ously the development of geometry and algebra led to the general concept of a linear space, while in analysis the basic forms of convergence for series of functions were identified: uniform, mean square, pointwise and so on. It turns out, moreover, that a specific type of convergence is associated with each linear function space, for example, uniform convergence in the case of the space of continuous functions on a closed interval. It was only comparatively recently that in this connection the general idea of a linear topological space (L TS)l was formed; here the algebraic structure is compatible with the topological structure in the sense that the basic operations (addition and multiplication by a scalar) are continuous.

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### Inhalt

 Classical Concrete Problems 6 12 Solution of Nonlinear Equations 7 13 Extremal Problems 10 15 Integration 11 16 Differential Equations 16 2 The Fourier Method and Related Questions 19 22 Heat Conduction 21 24 General Orthogonal Series 27
 23 Seminorms and Norms 111 24 The HahnBanach Theorem 112 25 Separating Hyperplanes 113 26 Nonnegative Linear Functionals 115 27 Ordered Linear Spaces 117 3 Linear Topology 119 32 Continuous Linear Functional 127 33 Complete Systems and Topological Bases 132

 25 Orthogonal Polynomials 28 26 The Power Moment Problem 29 27 Jacobian Matrices 34 28 The Trigonometric Moment Problem 35 29 The Fourier Integral 36 210 The Laplace Transform 39 211 The SturmLiouville Problem 45 212 The Schrodinger Operator on the Semiaxis 47 213 AlmostPeriodic Functions 51 3 Theory of Approximation 53 32 Chebyshev and Markov Systems 56 33 The ChebyshevMarkov Problem 58 34 The Lproblem of Moments 60 35 Interpolation and Quadrature Processes 61 36 Approximation in the Complex Plane 65 37 Quasianalytic Classes 69 4 Integral Equations 70 42 Fredholm and Volterra Equations 74 43 Fredholm Theory 76 44 HilbertSchmidt Theory 77 45 Equations with Difference Kernels 78 46 The RiemannHilbert Problem 82 Foundations and Methods 85 12 Homomorphisms and Linear Functional 88 13 The Algebraic Theory of the Index 93 14 Systems of Linear Equations 95 15 Algebraic Operators 97 16 General Principles of Summation of Series 100 17 Commutative Algebra 101 2 Convex Analysis 107 22 Convex Functionals 109
 34 Extreme Points of Compact Con vex Sets 138 35 Integration of VectorFunctions and Measures 140 36 wTopologies 144 37 Theory of Duality 147 38 Continuous Homomorphisms 153 39 Linearisation of Mappings 163 4 Theory of Operators 165 42 The Fixed Point Principle 174 43 Actions and Representations of Semigroups 178 44 The Spectrum and Resolvent of a Linear Operator 181 45 OneParameter Semigroups 186 46 Conjugation and Closure 192 47 Spectra and Extensions of Symmetric Operators 194 48 Spectral Theory of Selfadjoint Operators 197 49 Spectral Operators 206 410 Spectral Subspaces 207 411 Eigenvectors of Conservative and Dissipative Operators 211 412 Spectral Sets and Numerical Ranges 215 413 Complete Compact Operators 216 414 Triangular Decompositions 220 415 Functional Models 222 416 Indefinite Metric 227 417 Banach Algebras 232 5 Function Spaces 242 52 Generalised Functions 244 53 Families of Function Spaces 252 54 Operators on Function Spaces 255 Commentary on the Bibliography 261 Author Index 271 Subject Index 275 Urheberrecht

### Verweise auf dieses Buch

 Real AnalysisN. L. CarothersEingeschränkte Leseprobe - 2000