Offers graduate students, mathematicians, and scientists a clear presentation of fundamental topics of operator theory. Uses a systematic presentation, covering such important concepts as set theory, algebraic structures, topological structures, Banach spaces, and Hilbert spaces. DLC: Operator theory--Mathematics.
Author(s): Carlos S. Kubrusly
Edition: 1
Publisher: Birkhäuser Boston
Year: 2001
Language: English
Pages: 544
Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Preface......Page 8
Contents......Page 12
1.1 Background ......Page 16
1.2 Sets and Relations ......Page 18
1.3 Functions ......Page 20
1.4 Equivalence Relations......Page 22
1.5 Ordering ......Page 23
1.6 Lattices ......Page 25
1.7 Indexing ......Page 27
1.8 Cardinality ......Page 29
1.9 Remarks ......Page 36
Problems......Page 39
2.1 Linear Spaces......Page 52
2.2 Linear Manifolds ......Page 58
2.3 Linear Independence......Page 63
2.5 Linear Transformations ......Page 70
2.6 Isomorphisms ......Page 73
2.7 Isomorphic Equivalence ......Page 79
2.8 Direct Sum ......Page 81
2.9 Projections ......Page 85
Problems ......Page 90
3.1 Metric Spaces......Page 100
3.2 Convergence and Continuity ......Page 108
3.3 Open Sets and Topology ......Page 115
3.4 Equivalent Metrics and Homeomorphisms ......Page 121
3.5 Closed Sets and Closure ......Page 128
3.6 Dense Sets and Separable Spaces ......Page 134
3.7 Complete Spaces ......Page 142
3.8 Continuous Extension and Completion ......Page 149
3.9 The Baire Category Theorem ......Page 157
3.10 Compact Sets ......Page 163
3.11 Sequential Compactness ......Page 170
Problems ......Page 178
4.1 Normed Spaces ......Page 212
4.2 Examples ......Page 217
4.3 Subspaces and Quotient Spaces ......Page 224
4.4 Bounded Linear Transformations ......Page 230
4.5 The Open Mapping Theorem and Continuous Inverses......Page 238
4.6 Equivalence and Finite-Dimensional Spaces ......Page 245
4.7 Continuous Linear Extension and Completion ......Page 252
4.8 The Banach-Steinhaus Theorem and Operator Convergence ......Page 257
4.9 Compact Operators ......Page 265
4.10 The Hahn-Banach Theorem and Dual Spaces ......Page 273
Problems ......Page 284
5.1 Inner Product Spaces......Page 326
5.2 Examples ......Page 333
5.3 Orthogonality ......Page 338
5.4 Orthogonal Complement ......Page 343
5.5 Orthogonal Structure ......Page 350
5.6 Unitary Equivalence ......Page 354
5.7 Summability ......Page 358
5.8 Orthonormal Basis ......Page 367
5.9 The Fourier Series Theorem ......Page 374
5.10 Orthogonal Projection ......Page 382
5.11 The Riesz Representation Theorem and Weak Convergence ......Page 391
5.12 The Adjoint Operator ......Page 402
5.13 Self-Adjoint Operators ......Page 411
5.14 Square Root and Polar Decomposition ......Page 416
Problems ......Page 423
6 The Spectral Theorem ......Page 456
6.2 The Spectrum of an Operator ......Page 464
6.3 Spectral Radius ......Page 472
6.4 Numerical Radius ......Page 477
6.5 Examples of Spectra ......Page 481
6.6 The Spectrum of a Compact Operator ......Page 489
6.7 The Spectral Theorem for Compact Normal Operators......Page 495
6.8 A Glimpse at the Spectral Theorem for Normal Operators. ......Page 504
Problems. ......Page 511
References ......Page 524
Index ......Page 532
Back Cover......Page 544
