This book is an introductory text in functional analysis, aimed at the graduate student with a firm background in integration and measure theory. Unlike many modern treatments, this book begins with the particular and works its way to the more general, helping the student to develop an intuitive feel for the subject. For example, the author introduces the concept of a Banach space only after having introduced Hilbert spaces, and discussing their properties. The student will also appreciate the large number of examples and exercises which have been included.
Author(s): John B. Conway
Series: Graduate texts in mathematics 096
Edition: 2ed. (Corrected fourth printing)
Publisher: Springer
Year: 1997
Language: English
Pages: 414
Cover ......Page 1
Title ......Page 3
Preface ......Page 6
Preface to the Second Edition ......Page 10
Contents ......Page 11
§1. Elementary Properties and Examples ......Page 15
§2. Orthogonality ......Page 21
§3. The Riesz Representation Theorem ......Page 25
§4. Orthonormal Sets of Vectors and Bases ......Page 28
§5. Isomorphic Hilbert Spaces and the Fourier Transform for the Circle ......Page 33
§6. The Direct Sum of Hilbert Spaces ......Page 37
§1. Elementary Properties and Examples ......Page 40
§2. The Adjoint of an Operator ......Page 45
§3. Projections and Idempotents. Invariant and Reducing Subspaces ......Page 50
§4. Compact Operators ......Page 55
§5.* The Diagonalization of Compact Self-Adjoint Operators ......Page 60
§6.* An Application: Sturm-Liouville Systems ......Page 63
§7.* The Spectral Theorem and Functional Calculus for Compact Normal Operators ......Page 68
§8.* Unitary Equivalence for Compact Normal Operators ......Page 74
§1. Elementary Properties and Examples ......Page 77
§2. Linear Operators on Normed Spaces ......Page 81
§3. Finite Dimensional Normed Spaces ......Page 83
§4. Quotients and Products of Normed Spaces ......Page 84
§5. Linear Functionals ......Page 87
§6. The Hahn-Banach Theorem ......Page 91
§7.* An Application: Banach Limits ......Page 96
§8.* An Application: Runge's Theorem ......Page 97
§9.* An Application: Ordered Vector Spaces ......Page 100
§10. The Dual of a Quotient Space and a Subspace ......Page 102
§11. Reflexive Spaces ......Page 103
§12. The Open Mapping and Closed Graph Theorems ......Page 104
§13. Complemented Subspaces of a Banach Space ......Page 107
§14. The Principle of Uniform Boundedness ......Page 109
§1. Elementary Properties and Examples ......Page 113
§2. Metrizable and Normable Locally Convex Spaces ......Page 119
§3. Some Geometric Consequences of the Hahn-Banach Theorem ......Page 122
§4.* Some Examples of the Dual Space of a Locally Convex Space ......Page 128
§5.* Inductive Limits and the Space of Distributions ......Page 130
§1. Duality ......Page 138
§2. The Dual of a Subspace and a Quotient Space ......Page 142
§3. Alaoglu's Theorem ......Page 144
§4. Reflexivity Revisited ......Page 145
§5. Separability and Metrizability ......Page 148
§6.* An Application: The Stone-Cech Compactification ......Page 151
§7. The Krein-Milman Theorem ......Page 155
§8. An Application: The Stone-Weierstrass Theorem ......Page 159
§9.* The Schauder Fixed Point Theorem ......Page 163
§10.* The Ryll-Nardzewski Fixed Point Theorem ......Page 165
§11.* An Application: Haar Measure on a Compact Group ......Page 168
§12.* The Krein-Smulian Theorem ......Page 173
§13.* Weak Compactness ......Page 177
§1. The Adjoint of a Linear Operator ......Page 180
§2.* The Banach-Stone Theorem ......Page 185
§3. Compact Operators ......Page 187
§4. Invariant Subspaces ......Page 192
§5. Weakly Compact Operators ......Page 197
§1. Elementary Properties and Examples ......Page 201
§2. Ideals and Quotients ......Page 205
§3. The Spectrum ......Page 209
§4. The Riesz Functional Calculus ......Page 213
§5. Dependence of the Spectrum on the Algebra ......Page 219
§6. The Spectrum of a Linear Operator ......Page 222
§7. The Spectral Theory of a Compact Operator ......Page 228
§8. Abelian Banach Algebras ......Page 232
§9.* The Group Algebra of a Locally Compact Abelian Group ......Page 237
§1. Elementary Properties and Examples ......Page 246
§2. Abelian C*-Algebras and the Functional Calculus in C*-Algebras ......Page 250
§3. The Positive Elements in a C*-Algebra ......Page 254
§4.* Ideals and Quotients of C*-Algebras ......Page 259
§5.* Representations of C*-Algebras and the Gelfand-Naimark-Segal Construction ......Page 262
§1 Spectral Measures and Representations of Abelian C*-Algebras ......Page 269
§2. The Spectral TJieorem ......Page 276
§3. Star-Cyclic Normal Operators ......Page 282
§4. Some Applications of the Spectral Theorem ......Page 285
§5. Topologies on B(K) ......Page 288
§6. Commuting Operators ......Page 290
§7. Abelian von Neumann Algebras ......Page 295
§8. The Functional Calculus for Normal Operators: The Conclusion of the Saga ......Page 299
§9. Invariant Subspaces for Normal Operators ......Page 304
§10.Multiplicity Theory for Normal Operators: A Complete Set of Unitary Invariants ......Page 307
§1. Basic Properties and Examples ......Page 317
§2. Symmetric and Self-Adjoint Operators ......Page 322
§3. The Cayley Transform ......Page 330
§4. Unbounded Normal Operators and the Spectral Theorem ......Page 333
§5. Stone's Theorem ......Page 341
§6. The Fourier Transform and Differentiation ......Page 348
§7. Moments ......Page 357
§1.The Spectrum Revisited ......Page 361
§2.Fredholm Operators ......Page 363
§3.The Fredholm Index ......Page 366
§4.The Essential Spectrum ......Page 372
§5.The Components of FF ......Page 376
§6.A Finer Analysis of the Spectrum ......Page 377
§1. Linear Algebra ......Page 383
§2. Topology ......Page 385
APPENDIX B. The Dual of Lp(mu) ......Page 389
APPENDIX C. The Dual of C0(X) ......Page 392
Bibliography ......Page 398
List of Symbols ......Page 405
Index ......Page 409
Back cover ......Page 414
