Functional analysis: An introduction

Title Page......Page 1
Copyright......Page 2
Preface......Page 3
CONTENTS......Page 5
1.1. Basic Definitions......Page 10
1.2. Examples of Seminormed Linear Spaces......Page 15
1.3. Finite-Dimensional Normed Linear Spaces and a Theorem of Riesz......Page 21
1.4. Gauges and Seminorms......Page 28
1.5. Topology in Seminormed Linear Spaces......Page 32
1.6. Problems......Page 35
2.1. Topological Linear Spaces......Page 40
2.2. Finite-Dimensional Topological Linear Spaces......Page 44
2.3. Locally Convex Topological Linear Spaces......Page 46
2.4. Seminorms and Convex Balanced Absorbing Sets......Page 50
2.5. Frechet Spaces......Page 55
2.6. Problems......Page 58
3.1. Linear Transformations......Page 64
3.2. Some Basic Results Concerning Linear Transformations......Page 69
3.3. Some Basic Results Concerning Linear Functionals......Page 75
3.4. Problems......Page 83
4.1. The Hahn-Banach Theorem: Analytic Form......Page 89
4.2. Some Consequences of the Hahn-Banach Theorem......Page 94
4.3. The Hahn-Banach Theorem and Abelian Semigroups of Transformations......Page 99
4.4. Adjoint Transformations......Page 105
4.5. Separability of V......Page 107
4.6. Annihilators......Page 108
4.7. Ideals in L1(IR,dt)......Page 111
4.8. Continuous Linear Functionals on C([0,1])......Page 114
4.9. A Moment Problem......Page 118
4.10. Helly's Theorem......Page 120
4.11. Problems......Page 125
5.0. Introduction......Page 135
5.1. Linear Varieties and Hyperplanes......Page 136
5.2. The Hahn-Banach Theorem: Geometric Form......Page 141
5.3. Some Consequences of the Hahn-Banach Theorem Revisited......Page 147
5.4. Some Further Geometric Consequences of the Hahn-Banach Theorem......Page 149
5.5. Problems......Page 151
6.1. The Baire Category Theorem and Osgood's Theorem......Page 155
6.2. The Uniform Boundedness Theorem and the Banach-Steinhaus Theorem......Page 157
6.3. The Strong Operator Topology......Page 163
6.4. Local Membership in LgORdt)......Page 167
6.5. A Result in the Theory of Summability......Page 170
6.6. Divergent Fourier Series......Page 174
6.7. Problems......Page 178
7.1. Closed Mappings......Page 186
7.2. The Open Mapping Theorem......Page 189
7.3. The Closed Graph Theorem......Page 196
7.4. A Uniform Boundedness Theorem for Continuous Linear Functionals......Page 200
7.5. Some Results on Norms in C([0,1]).......Page 201
7.6. A Criterion for the Continuity of Linear Transformations on F- 2......Page 203
7.7. Separable Banach Spaces......Page 205
7.8. The Category of L1([-n,n],dt/2n) in Co(l)......Page 207
7.9. Problems......Page 208
8.1. Reflexive Spaces......Page 215
8.2. Uniform Convexity and Mil'man's Theorem......Page 223
8.3. Reflexivity of L P (X,S,N), 1 < p <......Page 232
8.4. Problems......Page 238
9.0. Introduction......Page 243
9.2. The Weak and Weak* Topologies......Page 248
9.3. Completeness in the Weak and Weak* Topologies......Page 254
9.4. The Banach-Alaoglu Theorem......Page 263
9.5. Banach Spaces as Spaces of Continuous Functions......Page 272
9.6. Banach Limits Revisited......Page 273
9.7. Fourier Series of Functions in L([-n,n],dt/2n),l < p <......Page 276
9.8. Multipliers......Page 281
9.9. Weak Compactness and Reflexivity......Page 284
9.10. A Theorem Concerning the Adjoint Transformation......Page 286
9.11. Problems......Page 291
10.1. The Bounded Weak* Topology......Page 299
10.2. The Krein-gmulian Theorem......Page 307
10.3. The Eberlein-S9mulian Theorem......Page 312
10.4. Problems......Page 321
11.0. Introduction......Page 325
11.1. Extreme Points and Extremal Sets......Page 326
11.2. The Krein-Mil'man Theorem......Page 331
11.3. L1( dt) Is Not a Dual Space......Page 335
11.4. The Stone-Weierstrass Theorem......Page 336
11.5. Helson Sets......Page 342
11.6. The Banach-Stone Theorem......Page 346
11.7. Problems......Page 354
12.1. The Fixed Point Property......Page 358
12.2. Contraction Mappings......Page 361
12.3. The Markov-Kakutani Fixed Point Theorem......Page 363
12.4. The Picard Existence Theorem for Ordinary Differential Equations......Page 366
12.5. Haar Measure on Compact Abelian Topological Groups......Page 370
13.0. Introduction......Page 380
13.1. Basic Definitions and Results......Page 381
13.2. The Parallelogram and Polarization Identities......Page 385
13.3. Some Other General Properties of Hilbert Spaces......Page 391
13.4. The Orthogonal Decomposition Theorem and the Riesz Representation Theorem......Page 393
13.5. Orthogonal Projections......Page 401
13.6. Complete Orthonormal Sets......Page 405
13.7. Fourier Analysis in L^2([-T1,TT],dt/2n)......Page 418
13.8. Rademacher Functions......Page 426
13.9. The Hilbert Space Adjoint......Page 433
13.10. Self-adjoint and Unitary Transformations......Page 438
13.11. The Mean Ergodic Theorem......Page 443
13.12. A Theorem About H2......Page 450
13.13. Some Basic Results in Spectral Theory......Page 455
13.14. Some Spectral Theory Results for Self-adjoint Transformations......Page 461
13.15. A Spectral Decomposition Theorem, for Compact Self-adjoint Transformations......Page 467
13.16. Problems......Page 479
REFERENCES......Page 493
INDEX......Page 498