Introductory Functional Analysis with Applications

✍ Scribed by Erwin Kreyszig

• Publisher: Wiley
• Year: 1989
• Tongue: English
• Leaves: 703
• Series: Wiley Classics Library
• Edition: 1
• Category: Library

✦ Synopsis

"Provides avenues for applying functional analysis to the practical study of natural sciences as well as mathematics. Contains worked problems on Hilbert space theory and on Banach spaces and emphasizes concepts, principles, methods and major applications of functional analysis."

✦ Table of Contents

INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS.......................................... 1
PREFACE................................................................................. 4
CONTENTS................................................................................ 8
NOTATIONS............................................................................... 12
METRIC SPACES........................................................................... 16
1.1 Metric Space.................................................................... 17
1.2 Further Examples of Metric Spaces............................................... 24
1.3 Open Set, Closed Set, Neighborhood.............................................. 33
1.4 Convergence, Cauchy Sequence, Completeness...................................... 40
1.5 Examples. Completeness Proofs................................................... 48
1.6 Completion of Metric Spaces..................................................... 56
NORMED SPACES. BANACH SPACES............................................................ 64
2.1 Vector Space.................................................................... 65
2.2 Normed Space. Banach Space...................................................... 73
2.3 Further Properties of Normed Spaces............................................. 82
2.4 Finite Dimensional Normed Spaces and Subspaces.................................. 87
2.5 Compactness and Finite Dimension................................................ 92
2.6 Linear Operators................................................................ 97
2.7 Bounded and Continuous Linear Operators.........................................106
2.8 Linear Functionals..............................................................119
2.9 Linear Operators and Functionals on Finite Dimensional Spaces...................127
2.10 Normed Spaces of Operators. Dual Space.........................................132
INNER PRODUCT SPACES. HILBERT SPACES....................................................142
3.1 Inner Product Space. Hilbert Space..............................................143
3.2 Further Properties of Inner Product Spaces......................................152
3.3 Orthogonal Complements and Direct Sums..........................................157
3.4 Orthonormal Sets and Sequences..................................................166
3.5 Series Related to Orthonormal Sequences and Sets................................175
3.6 Total Orthonormal Sets and Sequences............................................183
3.7 Legendre, Hermite and Laguerre Polynomials......................................191
3.8 Representation of Functionals on Hilbert Spaces.................................203
3.9 Hilbert-Adjoint Operator........................................................210
3.10 Self-Adjoint, Unitary and Normal Operators.....................................216
FUNDAMENTAL THEOREMS FOR NORMED AND BANACH SPACES.......................................224
4.1 Zorn's Lemma....................................................................225
4.2 Hahn-Banach Theorem.............................................................228
4.3 Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces.................233
4.4 Application to Bounded Linear Functionals on C[a, b]............................240
4.5 Adjoint Operator................................................................246
4.6 Reflexive Spaces................................................................254
4.7 Category Theorem. Uniform Boundedness Theorem...................................261
4.8 Strong and Weak Convergence.....................................................271
4.9 Convergence of Sequences of Operators and Functionals...........................278
4.10 Application to Summability of Sequences........................................285
4.11 Numerical Integration and Weak* Convergence....................................291
4.12 Open Mapping Theorem...........................................................300
4.13 Closed Linear Operators. Closed Graph Theorem..................................307
FURTHER APPLICATIONS: BANACH FIXED POINT THEOREM........................................314
5.1 Banach Fixed Point Theorem......................................................315
5.2 Application of Banach's Theorem to Linear Equations.............................322
5.3 Application of Banach's Theorem to Differential Equations.......................330
5.4 Application of Banach's Theorem to Integral Equations...........................334
FURTHER APPLICATIONS: APPROXIMATION THEORY..............................................342
6.1 Approximation in Normed Spaces..................................................342
6.2 Uniqueness, Strict Convexity....................................................345
6.3 Uniform Approximation...........................................................351
6.4 Chebyshev Polynomials...........................................................360
6.5 Approximation in Hilbert Space..................................................367
6.6 Splines.........................................................................372
SPECTRAL THEORY OF LINEAR OPERATORS IN NORMED SPACES....................................378
7.1 Spectral Theory in Finite Dimensional Normed Spaces.............................379
7.2 Basic Concepts..................................................................385
7.3 Spectral Properties of Bounded Linear Operators.................................390
7.4 Further Properties of Resolvent and Spectrum....................................395
7.5 Use of Complex Analysis IN Spectral Theory......................................401
7.6 Banach Algebras.................................................................409
7.7 Further Properties of Banach Algebras...........................................413
COMPACT LINEAR OPERATORS ON NORMED SPACES AND THEIR SPECTRUM............................420
8.1 Compact Linear Operators on Normed Spaces.......................................421
8.2 Further Properties of Compact Linear Operators..................................427
8.3 Spectral Properties of Compact Linear Operators on Normed Spaces................435
8.4 Further Spectral Properties of Compact Linear ()perators........................443
8.5 Operator Equations Involving Compact Linear Operators...........................451
8.6 Further Theorems of Fredholm Type...............................................457
8.7 Fredholm Alternative............................................................466
SPECTRAL THEORY OF BOUNDED SELF-ADJOINT LINEAR OPERATORS................................474
9.1 Spectral Properties of Bounded Self-Adjoint Linear Operators....................475
9.2 Further Spectral Properties of Bounded Self-Adjoint Linear Operators............480
9.3 Positive Operators..............................................................485
9.4 Square Roots of a Positive Operator.............................................491
9.5 Projection Operators............................................................495
9.6 Further Properties of Projections...............................................501
9.7 Spectral Family.................................................................507
9.8 Spectral Family of a Bounded Self-Adjoint Linear Operator.......................512
9.9 Spectral Representation of Bounded Self-Adjoint Linear Operators................520
9.10 Extension of the Spectral Theorem to Continuous Functions......................528
9.11 Properties of the Spectral Family of a Bounded Self-Adjoint Linear Operator....531
UNBOUNDED LINEAR OPERATORS IN HILBERT SPACE.............................................538
10.1 Unbounded Linear Operators and their Hilbert-Adjoint Operators.................539
10.2 Hilbert-Adjoint Operators, Symmetric and Self-Adjoint Linear Operators.........545
10.3 Closed Linear Operators and Closures...........................................550
10.4 Spectral Properties of Self-Adjoint. Linear Operators..........................556
10.5 Spectral Representation of Unitary Operators...................................561
10.6 Spectral Representation of Self-Adjoint Linear operators.......................571
10.7 Multiplication Operator and Differentiation Operator...........................577
UNBOUNDED LINEAR OPERATORS IN QUANTUM MECHANICS.........................................586
11.1 Basic Ideas. States, Observables, Position Operator............................587
11.2 Momentum Operator. Heisenberg Uncertainty Principle............................591
11.3 Time-Independent Schrodinger Equation..........................................598
11.4 Hamilton Operator..............................................................605
11.5 Time-Dependent SchrOdinger Equation............................................613
APPENDIX 1: SOME MATERIAL FOR REVIEW AND REFERENCE......................................624
A1.1 Sets...........................................................................624
A1.2 Mappings.......................................................................628
A1.3 Families.......................................................................632
A1.4 Equivalence Relations..........................................................633
A1.5 Compactness....................................................................633
A1.6 Supremum and Infimum...........................................................634
A1.7 Cauchy Convergence Criterion...................................................636
A1.8 Groups.........................................................................637
APPENDIX 2: ANSWERS TO ODD-NUMBERED PROBLEMS............................................638
Section 1.1.........................................................................638
Section 1.2.........................................................................638
Section 1.3.........................................................................639
Section 1.4.........................................................................640
Section 1.5.........................................................................640
Section 1.6.........................................................................642
Section 2.1.........................................................................642
Section 2.2.........................................................................642
Section 2.3.........................................................................643
Section 2.4.........................................................................644
Section 2.5.........................................................................644
Section 2.6.........................................................................645
Section 2.7.........................................................................645
Section 2.8.........................................................................647
Section 2.9.........................................................................647
Section 3.1.........................................................................648
Section 3.2.........................................................................649
Section 3.3.........................................................................650
Section 3.4.........................................................................650
Section 3.5.........................................................................651
Section 3.6.........................................................................652
Section 3.7.........................................................................652
Section 3.8.........................................................................653
Section 3.9.........................................................................654
Section 3.10........................................................................654
Section 4.1.........................................................................655
Section 4.2.........................................................................655
Section 4.3.........................................................................656
Section 4.5.........................................................................656
Section 4.6.........................................................................657
Section 4.7.........................................................................657
Section 4.8.........................................................................658
Section 4.9.........................................................................658
Section 4.10........................................................................658
Section 4.11........................................................................659
Section 4 .12.......................................................................659
Section 4.13........................................................................659
Section 5.1.........................................................................660
Section 5.2.........................................................................662
Section 5.3.........................................................................663
Section 5.4.........................................................................663
Section 6.2.........................................................................663
Section 6.3.........................................................................665
Section 6.4.........................................................................665
Section 6.5.........................................................................666
Section 6.6.........................................................................667
Section 7.1.........................................................................667
Section 7.2.........................................................................668
Section 7.3.........................................................................668
Section 7.4.........................................................................669
Section 7.5.........................................................................670
Section 7.6.........................................................................670
Section 7.7.........................................................................670
Section 8.1.........................................................................671
Section 8.2.........................................................................671
Section 8.3.........................................................................671
Section 8.4.........................................................................673
Section 8.6.........................................................................673
Section 8.7.........................................................................675
Section 9.1.........................................................................676
Section 9.2.........................................................................676
Section 9.3.........................................................................677
Section 9.4.........................................................................677
Section 9.5.........................................................................678
Section 9.6.........................................................................679
Section 9.8.........................................................................679
Section 9.9.........................................................................680
Section 9.11........................................................................681
Section 10.1........................................................................681
Section 10.2........................................................................682
Section 10.3........................................................................682
Section 10.4........................................................................683
Section 10.5........................................................................683
Section 10.6........................................................................684
Section 11.2........................................................................684
Section 11.3........................................................................686
Section 11.4........................................................................686
Section 11.5........................................................................688
APPENDIX 3: REFERENCES..................................................................690
INDEX...................................................................................696

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