Introduction to Functional Analysis

Title Page......Page 3
Copyright Information......Page 4
Dedication......Page 5
Preface......Page 6
Contents......Page 9
Introduction......Page 13
I The Abstract Approach to Linear Problems......Page 16
1.1 Abstract Linear Spaces......Page 17
1.2 Examples of Linear Spaces......Page 22
1.3 Linear Operators......Page 25
1.4 Linear Operators in Finite-Dimensional Spaces......Page 30
1.5 Other Examples of Linear Operators......Page 33
1.6 Direct Sums and Quotient Spaces......Page 40
1.7 Linear Functionals......Page 43
1.8 Linear Functionals in Finite-Dimensional Spaces......Page 47
1.9 Zorn\'s Lemma......Page 49
1.10 Extension Theorems for Linear Operators......Page 50
1.11 Hamel Bases......Page 53
1.12 The Transpose of a Linear Operator......Page 56
1.13 Annihilators, Ranges, and Null Spaces......Page 57
1.14 Conclusions......Page 61
II Topological Linear Spaces......Page 63
II.1 Normed Linear Spaces......Page 64
II.2 Examples of Normed Linear Spaces......Page 68
II.3 Finite-Dimensional Normed Linear Spaces......Page 74
II.4 Banach Spaces......Page 78
II.5 Quotient Spaces......Page 83
II.6 Inner-Product Spaces......Page 85
II.7 Hilbert Space......Page 98
II.8 Examples of Complete Orthonormal Sets......Page 103
II.9 Topological Linear Spaces......Page 106
II.10 Convex Sets......Page 112
II.11 Locally Convex Spaces......Page 117
II.12 Minkowski Functionals......Page 123
II.13 Metrizable Topological Linear Spaces......Page 127
III Linear Functionals and Weak Topologies......Page 133
III.1 Linear Varieties and Hyperplanes......Page 134
III.2 The Hahn-Banach Theorem......Page 137
III.3 The Conjugate of a Normed Linear Space......Page 146
III.4 The Second Conjugate Space......Page 151
III.5 Some Representations of Linear Functionals......Page 153
III.6 Weak Topologies for Linear Spaces......Page 168
III.7 Polar Sets and Annihilators......Page 172
III.8 Equicontinuity and S-topologies......Page 177
III.9 The Principle of Uniform Boundedness......Page 181
III.10 Weak Topologies for Normed Linear Spaces......Page 184
III.11 The Krein-Milman Theorem......Page 193
IV General Theorems on Linear Operators......Page 200
IV. 1 Spaces of Linear Operators......Page 201
IV.2 Integral Equations of the Second Kind......Page 208
IV.3 L^2 Kernels......Page 213
IV.4 Differential Equations and Integral Equations......Page 217
IV.5 Closed Linear Operators......Page 220
IV.6 Some Representations of Bounded Linear Operators......Page 231
IV.7 The M. Riesz Convexity Theorem......Page 236
IV.8 Coniugates of Linear Operators......Page 238
IV.9 Theorems About Continuous Inverses......Page 246
IV.10 The States of an Operator and Its Conjugate......Page 249
IV.11 Ad]oint Operators......Page 254
IV.12 Projections......Page 258
IV.13 Fredholm Operators......Page 265
V Spectral Analysis of Linear Operators......Page 276
V.1 Analytic Vector-Valued Functions......Page 277
V.2 The Resolvent Operator......Page 284
V.3 The Spectrum of a Bounded Linear Operator......Page 289
V.4 Subdivisions of the Spectrum......Page 294
V.5 Reducibility......Page 299
V.6 The Ascent and Descent of an Operator......Page 301
V.7 Compact Operators......Page 305
V.8 An Operational Calculus......Page 321
V.9 Spectral Sets. The Spectral Mapping Theorem......Page 332
V.10 Isolated Points of the Spectrum......Page 340
V.11 Operators with a Rational Resolvent......Page 348
Vl Spectral Analysis in Hilbert Space......Page 353
VI.1 Bilinear and Quadratic Forms......Page 354
VI.2 Symmetric Operators......Page 357
VI.3 Normal and Self-adjoint Operators......Page 361
VI.4 Compact Symmetric Operators......Page 365
VI.5 Symmetric Operators with Compact Resolvent......Page 373
VI.6 The Spectral Theorem for Bounded Self-adjoint Operators......Page 375
VI.7 Unitary Operators......Page 386
VI.8 Unbounded Self-adjoint Operators......Page 392
VII Banach Algebras......Page 398
VII.1 Examples of Banach Algebras......Page 399
VII.2 Spectral Theory in a Banach Algebra......Page 405
VII.3 Ideals and Homomorphisms......Page 412
VII.4 Commutative Banach Algebras......Page 416
VII.5 Applications and Extensions of the Gelland Theory......Page 427
VII.6 B*-algebras......Page 438
VII.7 The Spectral Theorem for a Normal Operator......Page 442
Bibliography......Page 457
List of Special Symbols......Page 467
Index......Page 471