泛函分析
2013-1
世界图书出版公司
Elias M. Stein
423
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《泛函分析（英文版）》在Princeton大学使用，同时在其它学校，比如UCLA等名校也在本科生教学中得到使用。其教学目的是，用统一的、联系的观点来把现代分析的“核心”内容教给本科生，力图使本科生的分析学课程能接上现代数学研究的脉络。
作者：（美国）斯坦恩（Stein E.M.） （美国）Rami Shakarchi
Foreword Preface Chapter 1. Lp Spaces and Banach Spaces 1 Lp spaces 1.1 The HSlder and Minkowski inequalities 1.2 Completeness of np 1.3 Further remarks 2 The case p = ∞ 3 Banach spaces 3.1 Examples 3.2 Linear functionals and the dual of a Banach space 4 The dual space of Lp when 1 < p < ∞ 5 More about linear functionals 5.1 Separation of convex sets 5.2 The Hahn-Banach Theorem 5.3 Some consequences 5.4 The problem of measure 6 Complex Lp and Banach spaces 7 Appendix: The dual of C(X) 7.1 The case of positive linear functionals 7.2 The main result 7.3 An extension 8 Exercises 9 Problems Chapter 2. Lp Spaces in Harmonic Analysis 1 Early Motivations 2 The Riesz interpolation theorem 2.1 Some examples 3 The Lp theory of the Hilbert transform 3.1 The L2 formalism 3.2 The Lp theorem 3.3 Proof of Theorem 3.2 4 The maximal function and weak-type estimates 4.1 The Lp inequality 5 The Hardy space H1r 5.1 Atomic decomposition of H1r 5.2 An alternative definition of H1r 5.3 Application to the Hilbert transform 6 The space H1r and maximal functions 6.1 The space BMO 7 Exercises 8 Problems Chapter 3. Distributions: Generalized Functions 1 Elementary properties 1.1 Definitions 1.2 Operations on distributions 1.3 Supports of distributions 1.4 Tempered distributions 1.5 Fourier transform 1.6 Distributions with point supports 2 Important examples of distributions 2.1 The Hilbert transform and pv(1/x) 2.2 Homogeneous distributions 2.3 Fundamental solutions 2.4 Fundamental solution to general partial differential equations with constant coefficients 2.5 Parametrices and regularity for elliptic equations 3 Caldeon-Zygmund distributions and Lp estimates 3.1 Defining properties 3.2 The Lp theory 4 Exercises 5 Problems Chapter 4. Applications of the Baire Category Theorem 1 The Balre category theorem 1.1 Continuity of the limit of a sequence of continuous functions 1.2 Continuous functions that are nowhere differentiable 2 The uniform boundedness principle 2.1 Divergence of Fourier series 3The open mapping theorem 3.1 Decay of Fourier coefficients of L1-functions 4 The closed graph theorem 4.1 Grothendieck's theorem on closed subspaces of Lp 5 Besicovitch sets 6 Exercises 7 Problems Chapter 5. Rudiments of Probability Theory 1 Bernoulli trials 1.1 Coin flips 1.2 The case N = ∞ 1.3 Behavior of as as N →∞, first results 1.4 Central limit theorem 1.5 Statement and proof of the theorem 1.6 Random series 1.7 Random Fourier series 1.8 Bernoulli trials 2 Sums of independent random variables 2.1 Law of large numbers and ergodic theorem 2.2 The role of martingales 2.3 The zero-one law 2.4 The central limit theorem 2.5 Random variables with values in Rd 2.6 Random walks 3 Exercises 4 Problems Chapter 6. An Introduction to Brownian Motion 1 The Framework 2 Technical Preliminaries 3 Construction of Brownian motion 4 Some further properties of Brownian motion 5 Stopping times and the strong Markov property 5.1 Stopping times and the Blumenthal zero-one law 5.2 The strong Markov property 5.3 Other forms of the strong Markov Property 6 Solution of the Dirichlet problem 7 Exercises 8 Problems Chapter 7. A Glimpse into Several Complex Variables 1 Elementary properties 2 Hartogs' phenomenon: an example 3 Hartogs' theorem: the inhomogeneous Cauchy-Riemann equations 4 A boundary version: the tangential Cauchy-Riemann equa-tions 5 The Levi form 6 A maximum principle 7 Approximation and extension theorems 8 Appendix: The upper half-space 8.1 Hardy space 8.2 Cauchy integral 8.3 Non-solvability 9 Exercises 10 Problems Chapter 8. Oscillatory Integrals in Fourier Analysis 1 An illustration 2 Oscillatory integrals 3 Fourier transform of surface-carried measures 4 Return to the averaging operator 5 Restriction theorems 5.1 Radial functions 5.2 The problem 5.3 The theorem 6 Application to some dispersion equations 6.1 The SchrSdinger equation 6.2 Another dispersion equation 6.3 The non-homogeneous SchrSdinger equation 6.4 A critical non-linear dispersion equation 7 A look back at the Radon transform 7.1 A variant of the Radon transform 7.2 Rotational curvature 7.3 Oscillatory integrals 7.4 Dyadic decomposition 7.5 Almost-orthogonal sums 7.6 Proof of Theorem 7.1 8 Counting lattice points 8.1 Averages of arithmetic functions 8.2 Poisson summation formula 8.3 Hyperbolic measure 8.4 Fourier transforms 8.5 A summation formula 9 Exercises 10 Problems Notes and References Bibliography Symbol Glossary Index
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《泛函分析(英文版)》的读者对象数学专业的本科生、研究生和相关专业的科研人员。
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