Lectures on Functional Analysis and Applications

Front Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Preface......Page 6
CONTENTS......Page 12
1.1.1. What is Functional Analysis......Page 22
1.1.2. Sets......Page 24
1.1.4. Sets Operations......Page 25
1.1.5. General Definition of a Function......Page 28
1.1.6. Images and Inverse Images of Sets. Inverse Mappings......Page 31
1.1.7. Properties of Inverse Mappings......Page 33
1.1.8. Relation of Order......Page 34
1.1.9. Choice Axiom......Page 36
Problems......Page 39
1.2.2. Definition of a Metric Space......Page 40
1.2.3. Open and Closed Sets. Vicinities of Points......Page 42
1.2.4. Convergence in a Metric Space......Page 44
1.2.5. Complete Metric Spaces......Page 45
1.2.7. Completion of a Metric Space......Page 46
Problems......Page 49
1.3.1. Definition of a Linear Space......Page 50
1.3.2. Linear Dependence and Independence of Vectors......Page 52
1.3.3. Subspaces. Linear Spans......Page 53
1.3.4. Quotient Spaces......Page 54
1.3.6. Normed Linear Spaces......Page 55
1.3.8. Scalar Product......Page 56
1.3.9. Hilbert Spaces......Page 60
1.3.10. Algebras......Page 61
1.3.11. Spaces of Continuous Functions and Spaces of Bounded Functions......Page 62
Problems......Page 64
1.4.1. Sets Operations in a Linear Space......Page 66
1.4.3 Linear Functions......Page 67
1.4.4. Linear Functionals......Page 68
1.4.5. Extension of a Linear Functional......Page 69
1.4.7. Hahn-Banach Theorem on Extension of a Linear Functional......Page 70
1.4.8. Kernel of a Linear Functional......Page 74
Problems......Page 75
2.1.2. Semi-Algebras of Sets......Page 76
2.1.3. Algebras of Sets......Page 78
2.1.4. Construction of a Semi-Algebra and an Algebra Induced by a Given Class of Sets......Page 79
2.1.6. Rings and Semi-Rings of Sets......Page 82
2.1.7. Monotone Classes of Sets......Page 83
2.1.8. Product of Two Spaces......Page 85
2.1.9. Product of a Set of Spaces......Page 86
Problems......Page 88
2.2.1. Additive Set Functions......Page 90
2.2.2. Continuous Set Functions......Page 93
2.2.3. General Properties of Measures......Page 94
2.2.4. Properties of Nonnegative Measures......Page 96
2.2.5. Representation of a Real Numeric Measure as a Difference of Nonnegative Measures......Page 101
2.2.6. Total Variation of a Measure......Page 104
Problems......Page 108
2.3.3. Exterior Measure......Page 110
2.3.4. Class of Lebesgue Measurable Sets......Page 115
2.3.6. Complete a-Algebra......Page 120
2.3.7. Coincidence of the a-Algebra of Lebesgue Measurable Sets with the Minimal Complete a-Algebra Containing the Semi-Algebra C......Page 121
2.3.8. Approximation Property of the Algebra Inducing a a- Algebra......Page 123
2.3.9. Coincidence of Classes of Sets C and C1......Page 126
2.3.10. General Extension of Numeric Measure Theorem......Page 127
2.3.11. Jordan Measure Extension......Page 128
Problems......Page 130
3.1.1. Definition of a Measurable Function......Page 134
3.1.2. Properties of Measurable Functions......Page 136
3.1.3. Simple and Elementary Functions......Page 137
3.1.4. Measurable Functions as Limits of Sequences of Elementary Functions......Page 138
3.1.5. Almost Everywhere Convergence......Page 140
3.1.7. Measures Induced by Measurable Functions......Page 143
Problems......Page 144
3.2.1. Convergence in Measure......Page 146
3.2.2. Almost Everywhere Convergence and Convergence in Measure......Page 148
3.2.3. Convergence in Measure of a Fundamental Sequence......Page 151
3.2.4. Almost Uniform Convergence......Page 152
3.2.5. Measurable Functions as Limits of Sequences of Simple Functions......Page 155
3.2.6. Property of Functions Measurable with Respect to 7-Algebra Induced by Another Function......Page 156
Problems......Page 158
3.3.2. Properties of an Integral of a Simple Function......Page 159
3.3.3. Integration of B-Space Valued Functions......Page 163
3.3.4. Correctness of Definition of an Integral......Page 164
3.3.5. Properties of Integrals......Page 167
3.3.6. Integrability of a Function whose Norm is Bounded by an Integrable Function......Page 170
3.3.7. Changing Variables......Page 174
3.3.9. Integrals of Numeric Functions by a B-Space-Valued Measure......Page 176
Problems......Page 177
3.4.1. Lebesgue Integral......Page 178
3.4.2. Lebesgue Integral by Lebesgue Measure......Page 180
3.4.3. Lebesgue and Riemann Integrals......Page 181
3.4.4. Riemann Improper Integral......Page 183
3.4.5. Lebesgue-Stieltjes and Riemann-Stielties Integrals......Page 186
3.4.6. Integrals of Functions with the Values in a B-Space in Lebesgue Measure......Page 188
Problems......Page 190
3.5.1. Monotone Convergence Theorem......Page 192
3.5.2. Termwise Integration of Series......Page 195
3.5.3. Fatou Lenuna......Page 196
3.5.4. Magorizable Sequence Theorem......Page 197
3.5.5. General Lebesgue Theorem on Passage to the Limit under the Integral Sign......Page 198
3.5.6. General Theorem on Termwise Integration of Series......Page 200
Problems......Page 202
3.6.1. Definitions......Page 205
3.6.2. Representation of a Measure as the Sum of Absolutely Continuous and Singular Measures......Page 207
3.6.3. Radon-Nikodym Theorem......Page 210
Problems......Page 213
3.7.1. Definition of a Lebesgue Space......Page 214
3.7.2. An Auxiliary Inequa- lity......Page 215
3.7.3. Norm in a Lebesgue Space......Page 216
3.7.4. Convergence in Mean......Page 218
3.7.5. Set of Simple Functions is Dense in a Lebesgue Space......Page 220
3.7.6. Hilbert Space of Scalar Functions with Integrable Square of Modulus......Page 222
3.7.7. Set of Continuous Functions is Dense in LP(X)......Page 223
3.7.8. Sobolev Spaces......Page 224
3.8.1. Measures in the Product of Two Spaces......Page 226
3.8.2. Fubini Theorem......Page 230
3.8.3. Multiple and Iterated Integrals......Page 232
3.8.4. Measures in Finite Products of Spaces......Page 233
3.8.5. Measures in Infinite Products of Spaces......Page 234
Problems......Page 239
4.1.1. Convergence and Continuity in Terms of Vicinities......Page 244
4.1.4. Internal Points, Adherent Points and Limit Points......Page 247
4.1.5. Bases and Subbases......Page 249
4.1.6. Tychonoff Product of Topological Spaces......Page 251
Problems......Page 252
4.2.1. Separability Axioms......Page 253
4.2.3. Countability Axioms......Page 257
4.2.4. Dense Sets. Separable Spaces......Page 258
Problems......Page 260
4.3.1. Convergence of a Sequence......Page 261
4.3.2. Continuity of a Function......Page 262
4.3.3. A Way to Assign a Topology......Page 264
4.3.4. Measurable Topological Spaces......Page 265
Problems......Page 266
4.4.1. Compact Sets and Spaces......Page 267
4.4.2. Centred System of Closed Sets......Page 268
4.4.3. Properties of Compact Sets and Spaces......Page 269
4.4.5. Continuous Mappings of Compact Sets......Page 273
4.5.1. Totally Bounded Sets in a Metric Space......Page 274
4.5.2 Properties of Compact Sets in Metric Spaces......Page 276
4.5.3. Continuous Functions on a Compact......Page 278
4.5.4. A Criterion of Precompactness of a Set of Continuous Functions......Page 279
4.6.1. Definition of a Topological Linear Space......Page 283
4.6.2. Fundamental Sequences......Page 284
4.6.4. Ways to Assign Topology in a Linear Space......Page 285
4.6.5. Continuous Linear Functions in Topological Linear Spaces......Page 290
4.6.7. Bounded Linear Functions in Normed Spaces......Page 291
4.6.8. Norm of a Linear Function......Page 292
4.6.9. Hahn-Banach Theorem for Normed Linear Spaces......Page 293
Problems......Page 295
4.7.1. Definition of a Weak Topology......Page 298
4.7.2. Weak Convergence......Page 299
4.7.3. Weakly Continuous Functions......Page 300
4.7.4. Weakly Measurable Functions......Page 301
4.7.5. Coincidence of Measurability and Weak Measurability of Functions with Values in Separable B-Space......Page 303
Problems......Page 304
5.1.1. Operations on Operators and Functionals......Page 306
5.1.2. Spaces of Linear Operators and Functionals......Page 311
5.1.3. Dual Spaces......Page 313
5.1.4. Topologies in a Space of Bounded Linear Operators......Page 316
Problems......Page 318
5.2.1. Definition of a Weak Integral......Page 319
5.2.2. Properties of a Weak Integral......Page 321
5.2.3. Relation between a Weak Integral and a Strong One......Page 323
5.2.4. Passage to the Limit under a Weak Integral Sign......Page 326
5.2.5. Fubini Theorem......Page 327
Problems......Page 328
5.3.1. Notion of a Generalized Function......Page 330
5.3.3. Space of Test Functions......Page 331
5.3.4. Definition of a Generalized Function......Page 335
5.3.5. Space of Generalized Functions......Page 337
5.3.7. Local Properties of Generalized Functions......Page 339
5.3.8. Change of Variables in Generalized Functions......Page 341
5.3.9. Differentiation of Generalized Functions......Page 342
5.3.10. Series of Generalized Functions......Page 344
5.3.11. Product of a Generalized Function and a Test Function......Page 347
5.3.12. Primitive of a Generalized Function......Page 348
5.3.13. Representation of the b-Function by Fourier Integral......Page 350
5.3.14. Representation of Derivatives of the b-Functions by the Fourier Integral......Page 352
5.3.15. Products of the b-Functions and their Derivatives......Page 354
Problems......Page 355
5.4.1. Dual Space of the Space of Continuous Functions......Page 359
5.4.3. Dual Spaces of the Spaces of Differentiable Functions......Page 370
5.4.4. The b-Function and its Derivatives as Continuous Linear Functionals on the Spaces of the Differentiable Functions......Page 373
5.4.5. Dual Spaces of Lebesgue Spaces......Page 382
Problems......Page 386
6.1.1. Closed Operators......Page 390
6.1.2. Commutativity of an Integral with a Linear Operator......Page 392
6.1.3. Adjoint Operators......Page 394
6.1.4. Existence of Adjoint Operator......Page 403
6.1.5. Positive Operators......Page 406
6.1.6. Isometric Operators......Page 407
6.1.7. Unitary Operators......Page 408
6.1.9. Banach-Steinhaus Theorem......Page 409
6.1.10. Bounded Linear Operators in a Normed Space......Page 412
6.1.11. Banach Theorem about Inverse Operator......Page 414
Problems......Page 416
6.2.2. Derivatives of B-Space Valued Functions with Respect to a Real Variable......Page 418
6.2.3. Integrals of B-Space-Valued Functions by a Real Variable......Page 420
6.2.4. Differential Equations in B-Spaces......Page 421
6.2.5. Integral Equations......Page 425
6.2.7. Existence of the Unique Fixed Point of an Operator......Page 426
6.2.8. Existence of the Unique Solution of an Integral Equation......Page 430
6.2.9. Existence of the Unique Solution of a Differential Equation......Page 432
6.2.11. Existence of the Unique Solution of a Fredhohn Linear Integral Equation......Page 434
Problems......Page 436
6.3.1. Eigenvalues......Page 438
6.3.2. Resolvent and Spectrum of an Operator......Page 439
6.3.3. Properties of Resolvents......Page 442
6.3.4. Properties of the Spectrum of a Linear Operator in a B-Space......Page 443
6.3.5. Spectrum of an Unitary Operator......Page 445
Problems......Page 446
7.1.1. Orthogonal Subspaces......Page 450
7.1.2. Linear Functionals......Page 453
7.1.3. Extension of a Linear Functional......Page 454
7.2.1. Bounded Linear Operators......Page 456
7.2.2. Isometric and Unitary Operators in H-Spaces......Page 459
7.2.3. Fourier-Plancherel Operators......Page 460
7.2.4. Unbounded Linear Operators......Page 465
Problems......Page 469
7.3.1. Self-Adjoint and Symmetric Operators......Page 470
7.3.2. Formula for the Norm of a Self-Adjoint Operator......Page 471
7.3.3. Spectrum of a Self--Adjoint Operator......Page 472
Problems......Page 476
7.4.1. Projectors and their Properties......Page 477
7.4.2. Convergence of Sequences of Operators......Page 481
7.4.3. Sequences of the Projectors......Page 482
7.4.4. General Definition of a Projector......Page 483
Problems......Page 484
7.5.2. Orthogonal and Orthonormal Sequences......Page 485
7.5.3. Expansion of a Vector in Terms of Orthonormal Vectors......Page 486
7.5.4. Complete Sequences of Vectors and Bases......Page 488
7.5.5. Representation of Functions by Series......Page 489
7.5.6. Conditions of Existence of a Basis in a If-Space......Page 500
7.5.7. Biorthogonal and Biorthonormal Sequences......Page 501
7.5.8. Expansion of a Vector in Terms of a Basis Generated by a Biorthonormal Sequence......Page 504
Problems......Page 506
7.6.2. Properties of Compact Operators......Page 511
7.6.3. Spectrum of a Compact Operator......Page 515
7.6.4 Normal Operators......Page 521
7.6.5. Operators with the Schmidt Norm......Page 522
7.6.6. Hilbert-Schmidt Operators......Page 525
7.6.7. Trace Type Operators......Page 529
7.6.8. Fredholm Linear Integral Equations......Page 531
Problems......Page 534
8.1.1. Existence of Eigenvalues......Page 536
8.1.2. Spectral Decomposition......Page 538
8.1.3. Linear Fredholm Integral Equations with Symmetric Kernel......Page 542
8.1.4. Method for Solving Integral Equations of Some Class......Page 545
8.1.5. Integral Representation of an Operator......Page 558
8.1.6. Decomposition of the Identity......Page 560
Problems......Page 562
8.2.1. Integral of a Simple Function over an Operator-Valued Measure......Page 563
8.2.2. Integral of a Measurable Function over an Operator-Valued Measure......Page 565
8.2.3. Adjoint Operator......Page 568
8.2.4. Commutativity of Operators......Page 572
8.2.5. Normal Operators......Page 575
8.2.6. Inverse Operators......Page 576
8.2.7. Spectrum......Page 578
8.2.8. Canonical Forms of Operators......Page 579
Problems......Page 583
8.3.1. Operator Polynomials and Series......Page 585
8.3.2. Polynomials of an Unitary Operator......Page 586
8.3.3. Functions of an Unitary Operator......Page 588
8.3.4. Cayley Transform......Page 593
8.3.5. Functions of a Self-Adjoint Operator......Page 594
8.3.6. Spectral Measure of a Self-Adjoint Operator......Page 595
8.4.1. Integral Representation of Operator Functions of Class C......Page 597
8.4.2. Spectral Decomposition of a Self-Adjoint Operator......Page 598
8.4.3. Spectral Decomposition of an Unitary Operator......Page 602
8.4.4. Spectral Decomposition of the Group of Unitary Operators......Page 606
8.4.5. Spectral Decomposition of a Normal Operator......Page 609
Problems......Page 611
9.1.1. Strong Differentials and Derivatives......Page 612
9.1.2. Differentiation of a Composite Function......Page 615
9.1.3. Finite Increments Formula......Page 616
9.1.4. Weak Differentials and Derivatives......Page 617
9.1.5. Relation between Weak Differentiability and Strong One......Page 618
9.1.6. Differentials and Derivatives of Higher Orders......Page 621
9.1.7. Taylor Formula and Series......Page 623
Problems......Page 627
9.2.1. Necessary Condition of an Extremum......Page 632
9.2.3. Sufficient Condition of an Extremum......Page 638
9.2.4. Conditional Extrema......Page 639
Problems......Page 642
9.3.1. Linear Differential Equations......Page 645
9.3.2. Evolution Operators......Page 648
9.3.3. Solution of Linear Differential Equations......Page 653
9.3.4. Nonlinear Differential Equations......Page 656
Problems......Page 659
10.1.1. General Comments about Approximate Methods......Page 666
10.1.2. Newton Method......Page 668
10.1.3. Modified Newton Method......Page 672
10.2.1. General Principles of Approximate Methods......Page 676
10.2.2. Convergence of Approximate Solutions to Exact Solution......Page 684
10.2.3. Rayleigh-Ritz Method......Page 689
10.2.4. Galerkin Method......Page 692
10.2.5. Finite Elements Method......Page 700
Problems......Page 703
10.3.1. Two Definitions of Correctness......Page 704
10.3.2. Regularization of Improper Problems......Page 707
10.3.3. Methods for Finding Regularizing Operator......Page 708
10.3.5. Generalization of Regularization Method......Page 716
Problems......Page 717
Table A.1.1. Operators, Weighting and Transfer Functions for Some Typical Linear Systems with the Lumped Parameters......Page 718
Table A.1.2. Operators, Weighting Green) and Transfer Functions for Some Typical Linear One--Dimensional Systems with the Distributed Parameters......Page 719
Table A.1.3. Typical Linear Discrete Two-Dimensional Signal and Image Processing Operators......Page 722
Table A.2.1. Definite Integrals......Page 727
Table A.2.2. Special Functions and Polynomials......Page 728
3. Laplace, Fourier, Sine and Cosine Transforms......Page 729
Table A.3.1. Laplace Transforms for Some Typical Functions......Page 730
Table A.3.2. Fourier Transforms for Some Typical Functions......Page 733
Table A.3.4. Sine Transforms for Some Typical Functions......Page 735
Table A.3.5. Discrete Laplace Transforms for Some Typical Functions......Page 737
References......Page 740
Subject Index......Page 744
Back Cover......Page 752