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