An Introduction to Metric Spaces and Fixed Point Theory

Preface ix

I Metric Spaces

1 Introduction 3

1.1 The real numbers R 3

1.2 Continuous mappings in R 5

1.3 The triangle inequality in R 7

1.4 The triangle inequality in R" 8

1.5 Brouwer's Fixed Point Theorem 10

Exercises 11

2 Metric Spaces 13

2.1 The metric topology 15

2.2 Examples of metric spaces 19

2.3 Completeness 26

2.4 Separability and connectedness 33

2.5 Metric convexity and convexity structures 35

Exercises 38

3 Metric Contraction Principles 41

3.1 Banach's Contraction Principle 41

3.2 Further extensions of Banach's Principle 46

3.3 The Caristi-Ekeland Principle 55

3.4 Equivalents of the Caristi-Ekeland Principle 58

3.5 Set-valued contractions 61

3.6 Generalized contractions 64

Exercises 67

4 Hyperconvex Spaces 71

4.1 Introduction 71

4.2 Hyperconvexity 77

4.3 Properties of hyperconvex spaces 80

4.4 A fixed point theorem 84

4.5 Intersections of hyperconvex spaces 87

4.6 Approximate fixed points 89

4.7 Isbell's hyperconvex hull 91

Exercises 98

5 "Normal" Structures in Metric Spaces 101

5.1 A fixed point theorem 101

5.2 Structure of the fixed point set 103

5.3 Uniform normal structure 106

5.4 Uniform relative normal structure 110

5.5 Quasi-normal structure 112

5.6 Stability and normal structure 115

5.7 Ultrametric spaces 116

5.8 Fixed point set structure—separable case 120

Exercises 123

II Banach Spaces

6 Banach Spaces: Introduction 127

6.1 The definition 127

6.2 Convexity 131

6.3 £2 revisited 132

6.4 The modulus of convexity 136

6.5 Uniform convexity of the tp spaces 138

6.6 The dual space: Hahn-Banach Theorem 142

6.7 The weak and weak* topologies 144

6.8 The spaces c, CQ, t\ and ^ 146

6.9 Some more general facts 148

6.10 The Schur property and £j 150

6.11 More on Schauder bases in Banach spaces 154

6.12 Uniform convexity and reflexivity 163

6.13 Banach lattices 165

Exercises 168

7 Continuous Mappings in Banach Spaces 171

7.1 Introduction 171

7.2 Brouwer's Theorem 173

7.3 Further comments on Brouwer's Theorem 176

7.4 Schauder's Theorem 179

7.5 Stability of Schauder's Theorem 180

7.6 Banach algebras: Stone Weierstrass Theorem 182

7.7 Leray-Schauder degree 183

7.8 Condensing mappings 187

7.9 Continuous mappings in hyperconvex spaces 191

Exercises 195

8 Metric Fixed Point Theory 197

8.1 Contraction mappings 197

8.2 Basic theorems for nonexpansive mappings 199

8.3 A closer look at ßë 205

8.4 Stability results in arbitrary spaces 207

8.5 The Goebel-Karlovitz Lemma 211

8.6 Orthogonal convexity 213

8.7 Structure of the fixed point set 215

8.8 Asymptotically regular mappings 219

8.9 Set-valued mappings 222

8.10 Fixed point theory in Banach lattices 225

Exercises 238

9 Banach Space Ultrapowers 243

9.1 Finite representability 243

9.2 Convergence of ultranets 248

9.3 The Banach space ultrapower X 249

9.4 Some properties of X 252

9.5 Extending mappings to X 255

9.6 Some fixed point theorems 257

9.7 Asymptotically nonexpansive mappings 262

9.8 The demiclosedness principle 263

9.9 Uniformly non-creasy spaces 264

Exercises 270

Appendix: Set Theory 273

A.l Mappings 273

A.2 Order relations and Zermelo's Theorem 274

A.3 Zorn's Lemma and the Axiom Of Choice 275

A.4 Nets and subnets 277

A.5 Tychonoff's Theorem 278

A.6 Cardinal numbers 280

A. 7 Ordinal numbers and transfinite induction 281

A.8 Zermelo's Fixed Point Theorem 284

A.9 A remark about constructive mathematics 286

Exercises 287

Bibliography 289

Index 301