INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONSPDF电子书下载

Chapter 1.Metric Spaces 1

1.1 Metric Space 2

1.2 Further Examples of Metric Spaces 9

1.3 Open Set, Closed Set, Neighborhood 17

1.4 Convergence, Cauchy Sequence, Completeness 25

1.5 Examples.Completeness Proofs 32

1.6 Completion of Metric Spaces 41

Chapter 2.Normed Spaces.Banach Spaces 49

2.1 Vector Space 50

2.2 Normed Space.Banach Space 58

2.3 Further Properties of Normed Spaces 67

2.4 Finite Dimensional Normed Spaces and Subspaces 72

2.5 Compactness and Finite Dimension 77

2.6 Linear Operators 82

2.7 Bounded and Continuous Linear Operators 91

2.8 Linear Functionals 103

2.9 Linear Operators and Functionals on Finite Dimen-sional Spaces 111

2.10 Normed Spaces of Operators.Dual Space 117

Chapter 3.Inner Product Spaces.Hilbert Spaces 127

3.1 Inner Product Space.Hilbert Space 128

3.2 Further Properties of Inner Product Spaces 136

3.3 Orthogonal Complements and Direct Sums 142

3.4 Orthonormal Sets and Sequences 151

3.5 Series Related to Orthonormal Sequences and Sets 160

3.6 Total Orthonormal Sets and Sequences 167

3.7 Legendre, Hermite and Laguerre Polynomials 175

3.8 Representation of Functionals on Hilbert Spaces 188

3.9 Hilbert-Adjoint Operator 195

3.10 Self-Adjoint, Unitary and Normal Operators 201

Chapter 4.Fundamental Theorems for Normed and Banach Spaces 209

4.1 Zorn’s Lemma 210

4.2 Hahn-Banach Theorem 213

4.3 Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces 218

4.4 Application to Bounded Linear Functionals on C[a, b] 225

4.5 Adjoint Operator 231

4.6 Reflexive Spaces 239

4.7 Cotegory Theorem.Uniform Boundedness Theorem 246

4.8 Strong and Weak Convergence 256

4.9 Convergence of Sequences of Operators and Functionals 263

4.10 Application to Summability of Sequences 269

4.11 Numerical Integration and Weak Convergence 276

4.12 Open Mapping Theorem 285

4.13 Closed Linear Operators.Closed Graph Theorem 291

Chapter 5.Further Applications: Banach Fixed Point Theorem 299

5.1 Banach Fixed Point Theorem 299

5.2 Application of Banach’s Theorem to Linear Equations 307

5.3 Applications of Banach’s Theorem to Differential Equations 314

5.4 Application of Banach’s Theorem to Integral Equations 319

Chapter 6.Further Applications: Approximation Theory 327

6.1 Approximation in Normed Spaces 327

6.2 Uniqueness, Strict Convexity 330

6.3 Uniform Approximation 336

6.4 Chebyshev Polynomials 345

6.5 Approximation in Hilbert Space 352

6.6 Splines 356

Chapter 7.Spectral Theory of Linear Operators in Normed Spaces 363

7.1 Spectral Theory in Finite Dimensional Normed Spaces 364

7.2 Basic Concepts 370

7.3 Spectral Properties of Bounded Linear Operators 374

7.4 Further Properties of Resolvent and Spectrum 379

7.5 Use of Complex Analysis in Spectral Theory 386

7.6 Banach Algebras 394

7.7 Further Properties of Banach Algebras 398

Chapter 8.Compact Linear Operators on Normed Spaces and Their Spectrum 405

8.1 Compact Linear Operators on Normed Spaces 405

8.2 Further Properties of Compact Linear Operators 412

8.3 Spectral Properties of Compact Linear Operators on Normed Spaces 419

8.4 Further Spectral Properties of Compact Linear Operators 428

8.5 Operator Equations Involving Compact Linear Operators 436

8.6 Further Theorems of Fredholm Type 442

8.7 Fredholm Alternative 451

Chapter9.Spectral Theory of Bounded Self-Adjoint Linear Operators 459

9.1 Spectral Properties of Bounded Self-Adjoint Linear Operators 460

9.2 Further Spectral Properties of Bounded Self-Adjoint Linear Operators 465

9.3 Positive Operators 469

9.4 Square Roots of a Positive Operator 476

9.5 Projection Operators 480

9.6 Further Properties of Projections 486

9.7 Spectral Family 492

9.8 Spectral Family of a Bounded Self-Adjoint Linear Operator 497

9.9 Spectral Representation of Bounded Self-Adjoint Linear Operators 505

9.10 Extension of the Spectral Theorem to Continuous Functions 512

9.11 Properties of the Spectral Family of a Bounded Self-Adjoint Linear Operator 516

Chapter 10.Unbounded Linear Operators in Hilbert Space 523

10.1 Unbounded Linear Operators and their Hilbert-Adjoint Operators 524

10.2 Hilbert-Adjoint Operators, Symmetric and Self-Adjoint Linear Operators 530

10.3 Closed Linear Operators and Closures 535

10.4 Spectral Properties of Self-Adjoint Linear Operators 541

10.5 Spectral Representation of Unitary Operators 546

10.6 Spectral Representation of Self-Adjoint Linear Operators 556

10.7 Multiplication Operator and Differentiation Operator 562

Chapter 11.Unbounded Linear Operators inffQuantum Mechanics 571

11.1 Basic Ideas.States, Observables, Position Operator 572

11.2 Momentum Operator.Heisenberg Uncertainty Principle 576

11.3 Time-lndependent Schrodinger Equation 583

11.4 Hamilton Operator 590

11.5 Time-Dependent Schrodinger Equation 598

Appendix 1.Some Material for Review and Reference 609

A1.1 Sets 609

A1.2 Mappings 613

A1.3 Families 617

A1.4 Equivalence Relations 618

A1.5 Compactness 618

A1.6 Supremum and Infimum 619

A1.7 Cauchy Convergence Criterion 620

A1.8 Groups 622

Appendix 2.Answers to Odd-Numbered Problems 623

Appendix 3.References 675

Index 681