《INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS》求取 ⇩

Chapter 1.Metric Spaces1

1.1 Metric Space2

1.2 Further Examples of Metric Spaces9

1.3 Open Set, Closed Set, Neighborhood17

1.4 Convergence, Cauchy Sequence, Completeness25

1.5 Examples.Completeness Proofs32

1.6 Completion of Metric Spaces41

Chapter 2.Normed Spaces.Banach Spaces49

2.1 Vector Space50

2.2 Normed Space.Banach Space58

2.3 Further Properties of Normed Spaces67

2.4 Finite Dimensional Normed Spaces and Subspaces72

2.5 Compactness and Finite Dimension77

2.6 Linear Operators82

2.7 Bounded and Continuous Linear Operators91

2.8 Linear Functionals103

2.9 Linear Operators and Functionals on Finite Dimen-sional Spaces111

2.10 Normed Spaces of Operators.Dual Space117

Chapter 3.Inner Product Spaces.Hilbert Spaces127

3.1 Inner Product Space.Hilbert Space128

3.2 Further Properties of Inner Product Spaces136

3.3 Orthogonal Complements and Direct Sums142

3.4 Orthonormal Sets and Sequences151

3.5 Series Related to Orthonormal Sequences and Sets160

3.6 Total Orthonormal Sets and Sequences167

3.7 Legendre, Hermite and Laguerre Polynomials175

3.8 Representation of Functionals on Hilbert Spaces188

3.9 Hilbert-Adjoint Operator195

3.10 Self-Adjoint, Unitary and Normal Operators201

Chapter 4.Fundamental Theorems for Normed and Banach Spaces209

4.1 Zorn’s Lemma210

4.2 Hahn-Banach Theorem213

4.3 Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces218

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

4.5 Adjoint Operator231

4.6 Reflexive Spaces239

4.7 Cotegory Theorem.Uniform Boundedness Theorem246

4.8 Strong and Weak Convergence256

4.9 Convergence of Sequences of Operators and Functionals263

4.10 Application to Summability of Sequences269

4.11 Numerical Integration and Weak Convergence276

4.12 Open Mapping Theorem285

4.13 Closed Linear Operators.Closed Graph Theorem291

Chapter 5.Further Applications: Banach Fixed Point Theorem299

5.1 Banach Fixed Point Theorem299

5.2 Application of Banach’s Theorem to Linear Equations307

5.3 Applications of Banach’s Theorem to Differential Equations314

5.4 Application of Banach’s Theorem to Integral Equations319

Chapter 6.Further Applications: Approximation Theory327

6.1 Approximation in Normed Spaces327

6.2 Uniqueness, Strict Convexity330

6.3 Uniform Approximation336

6.4 Chebyshev Polynomials345

6.5 Approximation in Hilbert Space352

6.6 Splines356

Chapter 7.Spectral Theory of Linear Operators in Normed Spaces363

7.1 Spectral Theory in Finite Dimensional Normed Spaces364

7.2 Basic Concepts370

7.3 Spectral Properties of Bounded Linear Operators374

7.4 Further Properties of Resolvent and Spectrum379

7.5 Use of Complex Analysis in Spectral Theory386

7.6 Banach Algebras394

7.7 Further Properties of Banach Algebras398

Chapter 8.Compact Linear Operators on Normed Spaces and Their Spectrum405

8.1 Compact Linear Operators on Normed Spaces405

8.2 Further Properties of Compact Linear Operators412

8.3 Spectral Properties of Compact Linear Operators on Normed Spaces419

8.4 Further Spectral Properties of Compact Linear Operators428

8.5 Operator Equations Involving Compact Linear Operators436

8.6 Further Theorems of Fredholm Type442

8.7 Fredholm Alternative451

Chapter9.Spectral Theory of Bounded Self-Adjoint Linear Operators459

9.1 Spectral Properties of Bounded Self-Adjoint Linear Operators460

9.2 Further Spectral Properties of Bounded Self-Adjoint Linear Operators465

9.3 Positive Operators469

9.4 Square Roots of a Positive Operator476

9.5 Projection Operators480

9.6 Further Properties of Projections486

9.7 Spectral Family492

9.8 Spectral Family of a Bounded Self-Adjoint Linear Operator497

9.9 Spectral Representation of Bounded Self-Adjoint Linear Operators505

9.10 Extension of the Spectral Theorem to Continuous Functions512

9.11 Properties of the Spectral Family of a Bounded Self-Adjoint Linear Operator516

Chapter 10.Unbounded Linear Operators in Hilbert Space523

10.1 Unbounded Linear Operators and their Hilbert-Adjoint Operators524

10.2 Hilbert-Adjoint Operators, Symmetric and Self-Adjoint Linear Operators530

10.3 Closed Linear Operators and Closures535

10.4 Spectral Properties of Self-Adjoint Linear Operators541

10.5 Spectral Representation of Unitary Operators546

10.6 Spectral Representation of Self-Adjoint Linear Operators556

10.7 Multiplication Operator and Differentiation Operator562

Chapter 11.Unbounded Linear Operators inffQuantum Mechanics571

11.1 Basic Ideas.States, Observables, Position Operator572

11.2 Momentum Operator.Heisenberg Uncertainty Principle576

11.3 Time-lndependent Schrodinger Equation583

11.4 Hamilton Operator590

11.5 Time-Dependent Schrodinger Equation598

Appendix 1.Some Material for Review and Reference609

A1.1 Sets609

A1.2 Mappings613

A1.3 Families617

A1.4 Equivalence Relations618

A1.5 Compactness618

A1.6 Supremum and Infimum619

A1.7 Cauchy Convergence Criterion620

A1.8 Groups622

Appendix 2.Answers to Odd-Numbered Problems623

Appendix 3.References675

Index681