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1 Bases and Basic Sequences 1

1.1 Schauder bases 1

1.2 Examples:Fourier series 6

1.3 Equivalence of bases and basic sequences 10

1.4 Bases and basic sequences:discussion 15

1.5 Constructing basic ？equences 19

1.6 The Eberlein-？mulian Theorem 23

Problems 25

2 The Classical Sequence Spaces 29

2.1 The isomorphic structure of the ep-spaces and c0 29

2.2 Complemented subspaces of ep（1≤p＜∞）and c0 33

2.3 The space e1 36

2.4 Convergence of series 38

2.5 Complementability of c0 44

Problems 48

3 Special Types of Bases 51

3.1 Unconditional bases 51

3.2 Boundedly-complete and shrinking bases 53

3.3 Nonreflexive spaces with unconditional bases 59

3.4 The James space ？ 62

3.5 A litmus test for unconditional bases 66

Problems 69

4 Banach Spaces of Continuous Functions 73

4.1 Basic properties 73

4.2 A characterization of real C(K)-spaces 75

4.3 Isometrically injective spaces 79

4.4 Spaces of continuous functions on uncountable compact metric spaces 87

4.5 Spaces of continuous functions on countable compact metric spaces 95

Problems 98

5 L1（μ）-Spaces and C(K)-Spaces 101

5.1 General remarks about L1（μ）-spaces 101

5.2 Weakly compact subsets of L1（μ） 103

5.3 Weak compactness in M(K) 112

5.4 The Danford-Pettis property 115

5.5 Weakly compact operators on C(K)-spaces 118

5.6 Subspaces of L1（μ）-spaces and C(K)-spaces 120

Problems 122

6 The Lp-Spaces for 1≤p＜∞ 125

6.1 Conditional expectations and the Haar basis 125

6.2 Averaging in Banach spaces 131

6.3 Properties of L1 142

6.4 Subspaces of Lp 148

Problems 161

7 Factorization Theory 165

7.1 Maurey-Nikishin factorization theorems 165

7.2 Subspaces of Lp for 1≤p＜2 173

7.3 Factoring through Hilbert spaces 180

7.4 The Kwapie？-Maurey theorems for type-2 spaces 187

Problems 191

8 Absolutely Summing Operators 195

8.1 Grothendieck's Inequality 196

8.2 Absolutely summing operators 205

8.3 Absolutely summing operators on L1（μ）-spaces 213

Problems 217

9 Perfectly Homogeneous Bases and Their Applications 221

9.1 Perfectly homogeneous bases 221

9.2 Symmetric bases 227

9.3 Uniqueness of unconditional basis 229

9.4 Complementation of block basic sequences 231

9.5 The existence of conditional bases 235

9.6 Greedy bases 240

Problems 244

10 ep-Subspaces of Banach Spaces 247

10.1 Ramsey theory 247

10.2 Rosenthal's e1 theorem 251

10.3 Tsirelson space 254

Problems 259

11 Finite Representability of ep-Spaces 263

11.1 Finite representability 263

11.2 The Principle of Local Reflexivity 272

11.3 Krivine's theorem 275

Problems 285

12 An Introduction to Local Theory 289

12.1 The John ellipsoid 289

12.2 The concentration of measure phenomenon 293

12.3 Dvoretzky's theorem 296

12.4 The complemented subspace problem 301

Problems 306

13 Important Examples of Banach Spaces 309

13.1 A generalization of the James space 309

13.2 Constructing Banach spaces via trees 314

13.3 Pelczy？ski's universal basis space 316

13.4 The James tree space 317

A Fundamental Notions 327

B Elementary Hilbert Space Theory 331

C Main Features of Finite-Dimensional Spaces 335

D Cornerstone Theorems of Functional Analysis 337

D.1 The Hahn-Banach Theorem 337

D.2 Baire's Theorem and its consequences 338

E Convex Sets and Extreme Points 341

F The Weak Topologies 343

G Weak Compactness of Sets and Operators 347

List of Symbols 349

References 353

Index 365