| Preface Introduction: To the Student 1 Metric Spaces, Normed Spaces, Inner Product Spaces 1.1 Basic Definitions and Theorems 1.2 Examples: Sequence Spaces and Function Spaces 1.3 A Discussion About Dimension Biography: Maurice Frechet Exercises for Chapter 1 2 The Topology of Metric Spaces 2.1 Open, Closed, and Compact Sets; the Heine-Borel and Ascoli-Arzela Theorems 2.2 Separability 2.3 Completeness: Banach and Hilbert Spaces Biography: David Hilbert Biography: Stefan Banach Exercises for Chapter 2 3 Measure and Integration 3.1 Probability Theory as Motivation 3.2 Lebesgue Measure on Euclidean Space Biography: Henri Lebesgue 3.3 Measurable and Lebesgue Integrable Functions on Euclidean Space 3.4 The Convergence Theorems 3.5 Comparison of the Lebesgue Integral with the Riemann Integral 3.6 General Measures and the Lebesgue LP-spaces: The Importance of Lebesgue's Ideas in Functional Analysis Biography: Frigyes Riesz Exercises for Chapter 3 4 Fourier Analysis in Hilbert Space 4.1 Orthonormal Sequences Biography: Joseph Fourier 4.2 Bessel's Inequality, Parseval's Theorem, and the Riesz-Fischer Theorem 4.3 A Return to Classical Fourier Analysis Exercises for Chapter 4 5 An Introduction to Abstract Linear Operator Theory 5.1 Basic Definitions and Examples 5.2 Boundedness and Operator Norms 5.3 Banach Algebras and Spectra; Compact Operators 5.4 An Introduction to the Invariant Subspace Problem Biography: Per Enflo 5.5 The Spectral Theorem for Compact Hermitian Operators Exercises for Chapter 5 6 Further Topics 6.1 The Classical Weierstrass Approximation Theorem and the Generalized Stone-Weierstrass Theorem Biography: Marshall Stone 6.2 The Baire Category Theorem with an Application to Real Analysis 6.3 Three Classical Theorems from Functional Analysis 6.4The Existence of a Nonmeasurable Set 6.5 Contraction Mappings 6.6 The Function Space C（[a, b]） as a Ring, and its Maximal Ideals 6.7 Hilbert Space Methods in Quantum Mechanics Biography: John von Nermann Exercises for Chapter 6 A Complex Numbers Exercises for Appendix A B Basic Set Theory Exercises for Appendix B References Index |
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