A First Course in Functional Analysis

Preface xi

1. Linear Spaces and Operators 1

1.1 Introduction 1

1.2 Linear Spaces 2

1.3 Linear Operators 5

1.4 Passage from Finite- to Infinite-Dimensional Spaces 7

Exercises 8

2. Normed Linear Spaces: The Basics 11

2.1 Metric Spaces 11

2.2 Norms 12

2.3 Space of Bounded Functions 18

2.4 Bounded Linear Operators 19

2.5 Completeness 21

2.6 Comparison of Norms 30

2.7 Quotient Spaces 31

2.8 Finite-Dimensional Normed Linear Spaces 34

2.9 Lᵖ Spaces 38

2.10 Direct Products and Sums 51

2.11 Schauder Bases 53

2.12 Fixed Points and Contraction Mappings 53

Exercises 54

3. Major Banach Space Theorems 59

3.1 Introduction 59

3.2 Baire Category Theorem 59

3.3 Open Mappings 61

3.4 Bounded Inverses 63

3.5 Closed Linear Operators 64

3.6 Uniform Boundedness Principle 66

Exercises 68

4. Hilbert Spaces 71

4.1 Introduction 71

4.2 Semi-Inner Products 72

4.3 Nearest Points and Convexity 77

4.4 Orthogonality 80

4.5 Linear Functionals on Hilbert Spaces 86

4.6 Linear Operators on Hilbert Spaces 88

4.7 Order Relation on Self-Adjoint Operators 97

Exercises 98

5. Hahn–Banach Theorem 103

5.1 Introduction 103

5.2 Basic Version of Hahn–Banach Theorem 104

5.3 Complex Version of Hahn–Banach Theorem 105

5.4 Application to Normed Linear Spaces 107

5.5 Geometric Versions of Hahn–Banach Theorem 108

Exercises 118

6. Duality 121

6.1 Examples of Dual Spaces 121

6.2 Adjoints 130

6.3 Double Duals and Reflexivity 133

6.4 Weak and Weak* Convergence 136

Exercises 140

7. Topological Linear Spaces 143

7.1 Review of General Topology 143

7.2 Topologies on Linear Spaces 148

7.3 Linear Functionals on Topological Linear Spaces 151

7.4 Weak Topology 153

7.5 Weak* Topology 156

7.6 Extreme Points and Krein–Milman Theorem 160

7.7 Operator Topologies 164

Exercises 164

8. The Spectrum 167

8.1 Introduction 167

8.2 Banach Algebras 169

8.3 General Properties of the Spectrum 170

8.4 Numerical Range 176

8.5 Spectrum of a Normal Operator 177

8.6 Functions of Operators 180

8.7 Brief Introduction to C_-Algebras 183

Exercises 184

9. Compact Operators 187

9.1 Introduction and Basic Definitions 187

9.2 Compactness Criteria in Metric Spaces 188

9.3 New Compact Operators from Old 192

9.4 Spectrum of a Compact Operator 194

9.5 Compact Self-Adjoint Operators on Hilbert Spaces 197

9.6 Invariant Subspaces 201

Exercises 203

10. Application to Integral and Differential Equations 205

10.1 Introduction 205

10.2 Integral Operators 206

10.3 Integral Equations 211

10.4 Second-Order Linear Differential Equations 214

10.5 Sturm–Liouville Problems 217

10.6 First-Order Differential Equations 223

Exercises 226

11. Spectral Theorem for Bounded, Self-Adjoint Operators 229

11.1 Introduction and Motivation 229

11.2 Spectral Decomposition 231

11.3 Extension of Functional Calculus 235

11.4 Multiplication Operators 240

Exercises 243

Appendix A Zorn’s Lemma 245

Appendix B Stone–Weierstrass Theorem 247

B.1 Basic Theorem 247

B.2 Nonunital Algebras 250

B.3 Complex Algebras 252

Appendix C Extended Real Numbers and Limit Points of Sequences 253

C.1 Extended Reals 253

C.2 Limit Points of Sequences 254

Appendix D Measure and Integration 257

D.1 Introduction and Notation 257

D.2 Basic Properties of Measures 258

D.3 Properties of Measurable Functions 259

D.4 Integral of a Nonnegative Function 261

D.5 Integral of an Extended Real-Valued Function 265

D.6 Integral of a Complex-Valued Function 267

D.7 Construction of Lebesgue Measure on R 267

D.8 Completeness of Measures 273

D.9 Signed and Complex Measures 274

D.10 Radon–Nikodym Derivatives 276

D.11 Product Measures 278

D.12 Riesz Representation Theorem 280

Appendix E Tychonoff’s Theorem 289

Symbols 293

References 297

Index 299