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CHAPTER Ⅰ Hilbert Spaces 1

1．Elementary Properties and Examples 1

2．Orthogonality 7

3．The Riesz Representation Theorem 11

4．Orthonormal Sets of Vectors and Bases 14

5．Isomorphic Hilbert Spaces and the Fourier Transform for the Circle 19

6．The Direct Sum of Hilbert Spaces 23

CHAPTER Ⅱ Operators on Hilbert Space 26

1．Elementary Properties and Examples 26

2．The Adioint of an Operator 31

3．Projections and Idempotents;Invariant and Reducing Subspaces 36

4．Compact Operators 41

5．The Diagonalization of Compact Self-Adjoint Operators 46

6．An Application:Sturm-Liouville Systems 49

7．The Spectral Theorem and Functional Calculus for Compact Normal Operators 54

8．Unitary Equivalence for Compact Normal Operators 60

CHAPTER Ⅲ Banach Spaces 63

1．Elementary Properties and Examples 63

2．Linear Operators on Normed Spaces 67

3．Finite Dimensional Normed Spaces 69

4．Quotients and Products of Normed Spaces 70

5．Linear Functionals 73

6．The Hahn-Banach Theorem 77

7．An Application:Banach Limits 82

8．An Application:Runge's Theorem 83

9．An Application:Ordered Vector Spaces 86

10．The Dual of a Quotient Space and a Subspace 88

11．Reflexive Spaces 89

12．The Open Mapping and Closed Graph Theorems 90

13．Complemented Subspaces of a Banach Space 93

14．The Principle of Uniform Boundedness 95

CHAPTER Ⅳ Locally Convex Spaces 99

1．Elementary Properties and Examples 99

2．Metrizable and Normable Locally Convex Spaces 105

3．Some Geometric Consequences of the Hahn-Banach Theorem 108

4．Some Examples of the Dual Space of a Locally Convex Space 114

5．Inductive Limits and the Space of Distributions 116

CHAPTER Ⅴ Weak Topologies 124

1．Duality 124

2．The Dual of a Subspace and a Quotient Space 128

3．Alaoglu's Theorem 130

4．Reflexivity Revisited 131

5．Separability and Metrizability 134

6．An Application:The Stone-？ech Compactification 137

7．The Krein-Milman Theorem 141

8．An Application:The Stone-Weierstrass Theorem 145

9．The Schauder Fixed Point Theorem 149

10．The Ryll-Nardzewski Fixed Point Theorem 151

11．An Application:Haar Measure on a Compact Group 154

12．The Krein-Smulian Theorem 159

13．Weak Compactness 163

CHAPTER Ⅵ Linear Operators on a Banach Space 166

1．The Adjoint of a Linear Operator 166

2．The Banach-Stone Theorem 171

3．Compact Operators 173

4．Invariant Subspaces 178

5．Weakly Compact Operators 183

CHAPTER Ⅶ Banach Algebras and Spectral Theory for Operators on a Banach Space 183

1．Elementary Properties and Examples 187

2．Ideals and Quotients 191

3．The Spectrum 195

4．The Riesz Functional Calculus 199

5．Dependence of the Spectrum on the Algebra 205

6．The Spectrum of a Linear Operator 208

7．The Spectral Theory of a Compact Operator 214

8．Abelian Banach Algebras 218

9．The Group Algebra of a Locally Compact Abelian Group 223

CHAPTER Ⅷ C*-Algebras 232

1．Elementary Properties and Examples 232

2．Abelian C*-Algebras and the Functional Calculus in C*-Algebras 236

3．The Positive Elements in a C*-Algebra 240

4．Ideals and Quotients of C*-Algebras 245

5．Representations of C*-Algebras and the Gelfand-Naimark-Segal Construction 248

CHAPTER Ⅸ Normal Operators on Hilbert Space 255

1．Spectral Measures and Representations of Abelian C*-Algebras 255

2．The Spectral Theorem 262

3．Star-Cyclic Normal Operators 268

4．Some Applications of the Spectral Theorem 271

5．Topologies on？(？) 274

6．Commuting Operators 276

7．Abelian von Neumann Algebras 281

8．The Functional Calculus for Normal Operators:The Conclusion of the Saga 285

9．Invariant Subspaces for Normal Operators 290

10．Multiplicity Theory for Normal Operators:A Complete Set of Unitary lnvariants 293

CHAPTER Ⅹ Unbounded Operators 303

1．Basic Properties and Examples 303

2．Symmetric and Self-Adjoint Operators 308

3．The Cayley Transform 316

4．Unbounded Normal Operators and the Spectral Theorem 319

5．Stone's Theorem 327

6．The Fourier Transform and Differentiation 334

7．Moments 343

CHAPTER Ⅺ Fredholm Theory 347

1．The Spectrum Revisited 347

2．Fredholm Operators 349

3．The Fredholm Index 352

4．The Essential Spectrum 358

5．The Components of？ 362

6．A Finer Analysis of the Spectrum 363

APPENDIX A Preliminaries 369

1．Linear Algebra 369

2．Topology 371

APENDIX B The Dual of Lp(μ) 375

APPENDIX C The Dual of C0(X) 378

Bibliography 384

List of Symbols 391

Index 395