Beginning Functional AnalysisSpringer Science & Business Media, 07.12.2001 - 197 Seiten This book is designed as a text for a first course on functional analysis for ad vanced undergraduates or for beginning graduate students. It can be used in the undergraduate curriculum for an honors seminar, or for a "capstone" course. It can also be used for self-study or independent study. The course prerequisites are few, but a certain degree of mathematical sophistication is required. A reader must have had the equivalent of a first real analysis course, as might be taught using [25] or [109], and a first linear algebra course. Knowledge of the Lebesgue integral is not a prerequisite. Throughout the book we use elementary facts about the complex numbers; these are gathered in Appendix A. In one spe cific place (Section 5.3) we require a few properties of analytic functions. These are usually taught in the first half of an undergraduate complex analysis course. Because we want this book to be accessible to students who have not taken a course on complex function theory, a complete description of the needed results is given. However, we do not prove these results. |
Inhalt
To the Student | 1 |
1 Metric Spaces Normed Spaces Inner Product Spaces | 3 |
11 Basic Definitions and Theorems | 4 |
Sequence Spaces and Function Spaces | 8 |
Biography Maurice Fréchet | 11 |
Exercises for Chapter 1 | 13 |
21 Open Closed and Compact Sets the HeineBorel and AscoliArzelà Theorems | 16 |
22 Separability | 22 |
Exercises for Chapter 4 | 88 |
51 Basic Definitions and Examples | 92 |
52 Boundedness and Operator Norms | 95 |
53 Banach Algebras and Spectra Compact Operators | 99 |
54 An Introduction to the Invariant Subspace Problem | 112 |
Per Enflo | 122 |
55 The Spectral Theorem for Compact Hermitian Operators | 124 |
Exercises for Chapter 5 | 129 |
Banach and Hilbert Spaces | 23 |
David Hilbert | 24 |
Stefan Banach | 28 |
Exercises for Chapter 2 | 29 |
Measure and Integration | 32 |
31 Probability Theory as Motivation | 33 |
32 Lebesgue Measure on Euclidean Space | 35 |
Henri Leon Lebesgue | 42 |
33 Measurable and Lebesgue Integrable Functions on Euclidean Space | 44 |
34 The Convergence Theorems | 50 |
35 Comparison of the Lebesgue Integral with the Riemann Integral | 54 |
The Importance of Lebesgues Ideas in Functional Analysis | 57 |
Frigyes Riesz | 68 |
Exercises for Chapter 3 | 69 |
41 Orthonormal Sequences | 74 |
Jean Baptiste Joseph Fourier | 79 |
42 Bessels Inequality Parsevals Theorem and the RieszFischer Theorem | 81 |
43 A Return to Classical Fourier Analysis | 85 |
Further Topics | 135 |
61 The Classical Weierstrass Approximation Theorem and the Generalized StoneWeierstrass Theorem | 136 |
Marshall Harvey Stone | 145 |
62 The Baire Category Theorem with an Application to Real Analysis | 146 |
63 Three Classical Theorems from Functional Analysis | 151 |
64 The Existence of a Nonmeasurable Set | 158 |
65 Contraction Mappings | 160 |
66 The Function Space Ca b as a Ring and its Maximal Ideals | 164 |
67 Hilbert Space Methods in Quantum Mechanics | 167 |
John von Neumann | 174 |
Exercises for Chapter 6 | 176 |
Complex Numbers | 181 |
Exercises for Appendix A | 183 |
Basic Set Theory | 185 |
Exercises for Appendix B | 186 |
| 187 | |
| 193 | |
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