Beginning functional analysis

Beginning Functional Analysis......Page 1
Preface......Page 3
Introduction: To the Student......Page 11
1 Metric Spaces, Norrned Spaces, Inner Product Spaces......Page 13
1.1 Basic Definitions and Theorems......Page 14
1.2 Examples: Sequence Spaces and Function Spaces......Page 18
1.3 A Discussion About Dimension......Page 21
Exercises for Chapter 1......Page 23
2.1 Open, Closed, and Compact Sets; the Heine-Bore1 and Ascoli-Arzel; Theorems......Page 26
2.2 Separability......Page 32
2.3 Completeness: Banach and Hilbert Spaces......Page 33
Exercises for Chapter 2......Page 39
3 Measure and Integration......Page 42
3.1 Probability Theory as Motivation......Page 43
3.2 Lebesgue Measure on Euclidean Space......Page 45
3.3 Measurable and Lebesgue Integrable Functions on Euclidean Space......Page 54
3.4 The Convergence Theorems......Page 60
3.5 Comparison of the Lebesgue Integral with the Riemann Integral......Page 64
3.6 General Measures and the Lebesgue Lp-spaces: The Importance of Lebesgue\'s Ideas in Functional Analysis......Page 67
Exercises for Chapter 3......Page 79
4.1 Orthonormal Sequences......Page 84
4.2 Bessel\'s Inequality, Parseval\'s Theorem, and the Riesz-Fischer Theorem......Page 91
4.3 A Return to Classical Fourier Analysis......Page 95
Exercises for Chapter 4......Page 98
5.1 Basic Definitions and Examples......Page 102
5.2 Boundedness and Operator Norms......Page 105
5.3 Banach Algebras and Spectra; Compact Operators......Page 109
5.4 An Introduction to the Invariant Subspace Problem......Page 122
5.5 The Spectral Theorem for Compact Hermitian Operators......Page 134
Exercises for Chapter 5......Page 139
6 Further Topics......Page 145
6.1 The Classical Weierstrass Approximation Theorem and the Generalized Stone-Weierstrass Theorem......Page 146
6.2 The Baire Category Theorem with an Application to Real Analysis......Page 156
6.3 Three Classical Theorems from Functional Analysis......Page 161
6.4 The Existence of a Nonmeasurable Set......Page 168
6.5 Contraction Mappings......Page 170
6.6 The Function Space C([a, b]) as a Ring, and its Maximal Ideals......Page 174
6.7 Hilbert Space Methods in Quantum Mechanics......Page 177
Exercises for Chapter 6......Page 186
Appendix A Complex Numbers......Page 191
Exercises for Appendix A......Page 193
Appendix B Basic Set Theory......Page 195
Exercises for Appendix B......Page 196
References......Page 197
Index......Page 203
Undergraduate Texts in Mathematics......Page 208