Elementary Functional Analysis [1 ed.] 0486318680, 9780486318684

Table of contents :
Preface vii

1 Basic Structures of Mathematical Analysis 1

1.1. Linear Spaces 2
1.2. Metric Spaces 25
1.3. Normed Linear Spaces 36
1.4. Hilbert Spaces 54
1.5. Approximation on a Compactum 68
1.6. Differentiation and Integration in a Normed Linear Space 83
1.7. Continuous Linear Operators 97
1.8. Normed Algebras 123
1.9. Spectral Properties of Linear Operators 131
Problems 141

2 Differential Equations 145

2.1. Definitions and Examples 145
2.2. The Fixed Point Theorem 160
2.3. Existence and Uniqueness of Solutions 163
2.4. Systems of Equations 169
2.5. Higher-Order Equations 172
2.6. Linear Equations and Systems 174
2.7. The Homogeneous Linear Equation 177
2.8. The Non-homogeneous Linear Equation 181
Problems 185

3 Space Curves 188

3.1. Basic Concepts 188
3.2. Higher Derivatives 193
3.3. Curvature 196
3.4. The Moving Basis 200
3.5. The Natural Equations 208
3.6. Helices 213
Problems 218

4 Orthogonal Expansions 220

4.1. Orthogonal Expansions in Hilbert Space 220
4.2. Trigonometric Fourier Series 226
4.3. Convergence of Fourier Series 232
4.4. Computations with Fourier Series 241
4.5. Divergent Fourier Series and Generalized Summation 258
4.6. Other Orthogonal Systems 264
Problems 270

5 The Fourier Transform 274

5.1. The Fourier Integral and Its Inversion 274
5.2. Further Properties of the Fourier Transform 280
5.3. Examples and Applications 293
5.4. The Laplace Transform 296
5.5. Quasi-Analytic Classes of Functions 307
Problems 317


Hints and Answers 320

Bibliography 326

Index 327

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