Functional analysis

Cover......Page 1
FUNCTIONAL ANALYSIS......Page 4
Library of Congress Catalog Card Number: 75-90855......Page 5
CONTENTS......Page 6
EDITOR'S FOREWORD TO THE RUSSIAN EDITION......Page 14
1. Concept of a linear system......Page 18
2. Linear dependence and independence......Page 19
1. Linear topological space......Page 20
2. Locally convex space......Page 22
3. Metric linear space......Page 23
4. Normed linear space......Page 24
5. Examples of normed linear spaces......Page 27
6. Completeness of metric spaces. Banach space......Page 31
7. Compact sets......Page 33
2. Continuous linear functionals......Page 36
3. Extension of continuous linear functionals......Page 37
4. Examples of linear functionals......Page 38
1. Duality of linear systems......Page 39
2. Conjugate space to a normed linear space.......Page 40
3. Weak and weak* topology......Page 44
4. Properties of a sphere in a conjugate Banach space......Page 45
5. Factor space and orthogonal complements......Page 46
6. Reflexive Banach spaces......Page 47
1. Bounded linear operators......Page 48
2. Examples of bounded linear operators. integral operators Interpolation theorems......Page 50
3. Convergence of a sequence of operators......Page 54
4. inverse operators......Page 55
6. Resolvent of a bounded linear operator. Spectrum......Page 56
7. Adjoint operator......Page 59
8. Completely continuous operators......Page 60
9. Operators with an everywhere dense domain of definition. Linear equations.......Page 64
10. Closed unbounded operato......Page 65
1. Completeness and minimality of a system of elements.......Page 68
2. Concept of a basis......Page 69
3. Criteria for bases......Page 71
4. Unconditional bases......Page 72
5. Stability of a basis......Page 73
2. Examples of Hilbert spaces......Page 74
3. Orthogonality. Projection onto a subspace......Page 76
4. Linear functionals......Page 77
6. Orthonormal systems......Page 78
1. Bounded linear operators. Adjoint operators. Bilinear forms......Page 80
2. Unitary operators......Page 82
3. Self-adjoint operators......Page 84
4. Self-adjoint completely continuous operators......Page 85
5. Completely continuous operators......Page 87
6. Projective operators......Page 90
1. Operations on seif-adjoint opera......Page 92
2. Resolution of the identity. The spectral function......Page 94
3. Functions of a seif-adjoint ope......Page 95
4. Unbounded seif-adjoint operators......Page 96
5. Spectrum of a seif-adjoint operator.......Page 98
6. Theory of perturbations......Page 99
7. Multiplicity of the spectrum of a seif-adjoint operator.......Page 102
8. Generalized eigenvectors.......Page 105
1. Concept of a symmetric operator, deficiency indices......Page 107
2. SeIf-adjoint extensions of symmetric operators......Page 108
3. SeIf-adjoint extensions of semi-bounded operators......Page 109
4. Dissipative extensions......Page 112
1. SeIf-adjoint differential expressions......Page 113
2. Regular case......Page 115
3. Singular case......Page 116
4. Criteria for self-adjointness of the operator Ao on (- \infinity, \infinity).......Page 118
6. Expansion in terms of eigenfunctions......Page 119
7. Examples......Page 122
8. Inverse Sturm-Liouville problem......Page 124
1. Self-adjoint elliptic differential expressions......Page 125
2. Minimal and maximal operators. L-harmonic functions......Page 126
3. Self-adjoint extensions corresponding to basic boundary value problenis.......Page 127
1. Hilbert scale and its properties......Page 130
2. Example of a Flilbert scale. The spaces W2......Page 131
3. Operators in a Hubert scale......Page 133
4. Theorems about traces......Page 134
2. Homogeneous equations with a constant operator......Page 136
3. Case of a Hilbert space......Page 138
5. Homogeneous equation with a variable operator......Page 139
1. Cauchy problem......Page 146
2. Uniformly correct Cauchy problem......Page 147
3. Generating operator and its resolvent......Page 149
4. Weakened Cauchy problem......Page 151
5. Abstract parabolic equation. Analytic semi-groups......Page 153
6. Reverse Cauchy problem......Page 154
7. Equations in a Hi/bert space......Page 156
8. Examples of well posed problems for partial differential equations......Page 159
9. Equations in a space with a basis. Continual integrals......Page 164
1. Homogeneous equation......Page 168
2. Case of an operator A (t) with a variable domain of definition......Page 170
3. Non-homogeneous equation......Page 171
4. Fractional powers of operators......Page 172
Introductory remarks......Page 175
1. Continuity and boundedness of an operator......Page 176
2. Differentiability of a nonlinear operator......Page 177
3. Integration of abstract functions......Page 179
4. Urysohn operator in the spaces C and Lp......Page 181
6. Hammerstein operator......Page 184
7. Derivatives of higher order......Page 185
8. Potential operators......Page 187
1. Method of successive approximations......Page 189
2. Principle of contractive mappings......Page 190
3. Uniqueness of a solution......Page 191
4. Equations with completely continuous operators. Schauder principle......Page 192
5. Use of the theory of completely continuous vector fields......Page 193
7. Transformation of equations......Page 196
8. Examples. Decomposition of operators.......Page 197
§ 3. Qualitative methods in the theory of branching of solutions......Page 200
2. Branch points......Page 201
3. Points of bifurcation, linearization principle......Page 203
4. Examples from mechanics.......Page 206
5. Equations with potential operators......Page 210
7. Equation of branching......Page 211
8. Construction of solutions in the form of a series......Page 212
1. Cone in a linear system......Page 215
2. Partially ordered spaces......Page 216
3. Vector lattices, minihedral cones......Page 217
4. K-spaces......Page 218
5. Cones in a Banach space......Page 219
6. Regular cones......Page 221
1. Positive functionals......Page 223
2. Extension of positive linear fun ctionals......Page 225
4. Bounded functionals on a cone......Page 226
1. Concept of a positive operator......Page 227
2. Affirmative eigenvalues......Page 228
3. Positive operators on a minihedral cone......Page 230
5. invariant functionals and eigenvectors of conjugate operators......Page 232
6. Inconsistent inequalities......Page 233
1. Basic concepts......Page 234
2. Existence of positive solutions......Page 235
3. Existence of a non-zero positive solution......Page 236
4. Concave operators......Page 237
5. Convergence of successive approximations......Page 238
2. Examples of normed rings......Page 239
4. Maximal ideals and multiplicative functionals......Page 242
5. Maximal ideal space......Page 244
7. Analytic functions on a ring......Page 245
9. Rings with involution......Page 247
1. Group rings......Page 248
2. The characters of a discrete group and maximal ideals of a group ring.......Page 250
3. Compact groups. Principle of duality......Page 252
4. Locally compact groups......Page 253
5. Fourier transforms......Page 254
6. Hypercomplex systems......Page 255
1. Regular rings......Page 256
3. The ring C(S) and its subrings......Page 258
Representations of algebraic systems......Page 260
3. Coordinates and impulses......Page 261
4. Energy operator. Schrôdinger equation......Page 263
5. Concrete quantum-mechanical systems......Page 265
6. Transition from quantum mechanics to classica/ mechanics.......Page 266
1. Criterion for self-adjointness......Page 269
2. Nature of the spectrum of a radial Schrädinger operator......Page 271
3. Nature of the spectrum of a one-dimensional Schrödinger operator......Page 272
4. Nature of the spectrum of a three-dimensional Schrodinger operator......Page 273
1. Exact solutions......Page 274
2. General properties of the solutions of the Schrödinger equation......Page 277
3. Quasi-classical approximation for solutions of the one-dimensional Schrödinger equation......Page 278
4. Calculation of eigen values in one-dimensional and radial symmetric cases......Page 281
5. Perturbation theory......Page 283
1. General information......Page 285
2. Theory of perturbations.......Page 286
3. Physical interpretation......Page 287
4. Quasi-classical asymptotics of the Green's function......Page 288
5. Passage to the limit as h—> 0......Page 290
6. Quasi-classical asyniptotics of a solution of the Dirac equation......Page 291
§ 5. Continuous spectrum of the energy operator and the problem of scattering......Page 294
1. Formulation of the problem......Page 295
2. Basis for the formulation of the problem and its solution......Page 296
3. Amplitude of scattering and its equation......Page 298
4. Case of spherical symmetry......Page 299
5. General case......Page 301
6. inverse problem of the theory of scattering......Page 302
1. Introductory remarks......Page 305
2. Notation......Page 306
3. Generalized functions......Page 307
4. Operations on generalized functions......Page 309
5. Differentiation and integration of generalized functions.......Page 310
6. Limit of a sequence of generalized functions......Page 312
7. Local properties of generalized functions......Page 314
8. Direct product of generalized functions......Page 315
9. Convolution of generalized functions......Page 316
10. Genera/form of generalized functions......Page 318
1. Regularization of divergent integrals......Page 319
2. Regularization of the functions x^2, x^3, x^-n and their linear combinations......Page 322
3. Regularization of functions with algebraic singularities......Page 325
4. Regularization on a finite segment.......Page 327
5. Regularization at infinity......Page 329
6. Non-canonical regularizations......Page 330
7. Generalized functions x^2+, x^2_, and functions which are analogous tothem as function of the parameter \lambda.......Page 333
8. Homogeneous generalized functions......Page 336
9. Table of derivatives of some generalized functions......Page 337
10. Differentiation and integration of arbitrary order......Page 338
11. Expression of some special functions in the form of derivatives of fractional order.......Page 339
1. The generalized function r^\lambda......Page 340
2. Generalized functions connected with quadratic forms......Page 343
3. Generalizedfunctions (P+iO)^\lambda and (P—iO)^\lambda.......Page 345
4. Generalized functions of the form......Page 346
5. Generalized functions on smooth surfaces......Page 348
1. The space S and generalized functions of exponential growth.......Page 351
2. Fourier transformation of generalizedfunctions of exponential growth......Page 352
3. Fourier transformation of arbitrary generalized functions......Page 354
4. Table of Fourier transforms of generalized functions of one variable.......Page 355
6. Positive definite generalized functions......Page 360
1. Radon transformation of test functions and its properties......Page 366
2. Radon transformation of generalized functions......Page 367
1. Fundamental solutions......Page 369
2. Fundamental solutions for some differential equations......Page 377
3. Construction of fundamental solutions for elliptic equations......Page 378
4. Fundamental solutions of homogeneous regular equations......Page 381
5. Fundamental solution of the Cauchy problem......Page 382
1. Generalizedfunctions of one complex variable......Page 385
2. Generalized functions of m complex variables......Page 389
BIBLIOGRAPHY......Page 394
INDEX OF LITERATURE ACCORDING TO CHAPTERS......Page 397