Functional Analysis I: Linear Functional Analysis

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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
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Real Analysis
N. L. Carothers
Eingeschränkte Leseprobe - 2000

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