.

0 (hodnocen0 x )

BK

Providence : American Mathematical Society, [2015]

xx, 789 stran : ilustrace ; 27 cm

ISBN 978-1-4704-1099-5 (vázáno)

Obsahuje bibliografii na stranách 713-763 a rejstříky

001453230

Contents // Preface to the Series xi // Preface to Part 1 xvii // Chapter 1. Preliminaries 1 // §1.1. Notation and Terminology 1 // §1.2. Metric Spaces 3 // §1.3. The Real Numbers 6 // §1.4. Orders 9 // §1.5. The Axiom of Choice and Zorn’s Lemma 11 // §1.6. Countability 14 // §1.7. Some Linear Algebra 18 // §1.8. Some Calculus 30 // Chapter 2. Topological Spaces 35 // §2.1. Lots of Definitions 37 // §2.2. Countability and Separation Properties 51 // §2.3. Compact Spaces 63 // §2.4. The Weierstrass Approximation Theorem and Bernstein // Polynomials 76 // §2.5. The Stone-Weierstrass Theorem 88 // §2.6. Nets 93 // §2.7. Product Topologies and Tychonoff’s Theorem 99 // §2.8. Quotient Topologies 103 // Chapter 3. A First Look at Hilbert Spaces and Fourier Series 107 // §3.1. Basic Inequalities 109 // §3.2. Convex Sets, Minima, and Orthogonal Complements 119 // §3.3. Dual Spaces and the Riesz Representation Theorem 122 // §3.4. Orthonormal Bases, Abstract Fourier Expansions, // and Gram-Schmidt 131 // §3.5. Classical Fourier Series 137 // §3.6. The Weak Topology 168 // §3.7. A First Look at Operators 174 // §3.8. Direct Sums and Tensor Products of Hilbert Spaces 176 // Chapter 4. Measure Theory 185 // §4.1. Riemann-Stieltjes Integrals 187 // §4.2. The Cantor Set, Function, and Measure 198 // §4.3. Bad Sets and Good Sets 205 // §4.4. Positive Functionals and Measures via L1 (A) 212 // §4.5. The Riesz-Markov Theorem 233 // §4.6. Convergence Theorems; Lp Spaces 240 // §4.7. Comparison of Measures 252 // §4.8. Duality for Banach Lattices; Hahn and Jordan // Decomposition 259 // §4.9. Duality for IS 270 // §4.10. Measures on Locally Compact and o-Compact Spaces 275 // §4.11. Product Measures and Fubini’s Theorem 281 // §4.12. Infinite Product Measures and Gaussian Processes 292 // §4.13. General Measure Theory 300 // §4.14. Measures on Polish Spaces 306 //

§4.15. Another Look at Functions of Bounded Variation 314 // §4.16. Bonus Section: Brownian Motion 319 // §4.17. Bonus Section: The Hausdorff Moment Problem 329 // §4.18. Bonus Section: Integration of Banach Space-Valued // Functions 337 // §4.19. Bonus Section: Haar Measure on cr-Compact Groups 342 // Chapter 5. Convexity and Banach Spaces 355 // §5.1. Some Preliminaries 357 // §5.2. Holder’s and Minkowski’s Inequalities: A Lightning Look 367 // §5.3. Convex Functions and Inequalities 373 // §5.4. The Baire Category Theorem and Applications 394 // §5.5. The Hahn-Banach Theorem 414 // §5.6. Bonus Section: The Hamburger Moment Problem 428 // §5.7. Weak Topologies and Locally Convex Spaces 436 // §5.8. The Banach-Alaoglu Theorem 446 // §5.9. Bonus Section: Minimizers in Potential Theory 447 // §5.10. Separating Hyperplane Theorems 454 // §5.11. The Krein-Milman Theorem 458 // §5.12. Bonus Section: Fixed Point Theorems and Applications 468 // Chapter 6. Tempered Distributions and the Fourier Transform 493 // §6.1. Countably Normed and Fréchet Spaces 496 // §6.2. Schwartz Space and Tempered Distributions 502 // §6.3. Periodic Distributions 520 // §6.4. Hermite Expansions 523 // §6.5. The Fourier Transform and Its Basic Properties 540 // §6.6. More Properties of Fourier Transform 548 // §6.7. Bonus Section: Riesz Products 576 // §6.8. Fourier Transforms of Powers and Uniqueness of Minimizers in Potential Theory 583 // §6.9. Constant Coefficient Partial Differential Equations 588 // 04Chapter 7. Bonus Chapter: Probability Basics 615 // §7.1. The Language of Probability 617 // §7.2. Borel-Cantelli Lemmas and the Laws of Large Numbers and of the Iterated Logarithm 632 // §7.3. Characteristic Functions and the Central Limit Theorem 648 // §7.4. Poisson Limits and Processes 660 // §7.5. Markov Chains 667 // Chapter 8. Bonus Chapter: Hausdorff Measure and Dimension 679 //

§8.1. The Carathéodory Construction 680 // §8.2. Hausdorff Measure and Dimension 687 // Chapter 9. Bonus Chapter: Inductive Limits and Ordinary // Distributions 705 // §9.1. Strict Inductive Limits 706 // §9.2. Ordinary Distributions and Other Examples of Strict // Inductive Limits 711 // Bibliography 713 // Symbol Index 765 // Subject Index 769 // Author Index 779 // Index of Capsule Biographies 789