آنالیز تابعی (تحصیلات تکمیلی در ریاضیات) (تحصیلات تکمیلی در ریاضیات، ۱۹۱)

Main subject categories: • Functional Analysis • Topology • Fredholm Theory • Spectral Theory • Unbounded Operators • Semigroups of OperatorsFunctional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, and physics. This book provides a comprehensive introduction to the field for graduate students and researchers.With 10 to 20 elaborate exercises at the end of each chapter, this book can be used as a text for a one-or-two-semester course on functional analysis for beginning graduate students.Prerequisites are first-year analysis and linear algebra, as well as some foundational material from the second-year courses on point set topology, complex analysis in one variable, and measure and integration. Cover......Page 1 Title page......Page 4 Contents......Page 6 Preface......Page 10 Introduction......Page 12 Chapter 1. Foundations......Page 16 1.1. Metric Spaces and Compact Sets......Page 17 1.2. Finite-Dimensional Banach Spaces......Page 32 1.3. The Dual Space......Page 40 1.4. Hilbert Spaces......Page 46 1.5. Banach Algebras......Page 50 1.6. The Baire Category Theorem......Page 55 1.7. Problems......Page 60 Chapter 2. Principles of Functional Analysis......Page 64 2.1. Uniform Boundedness......Page 65 2.2. Open Mappings and Closed Graphs......Page 69 2.3. Hahn–Banach and Convexity......Page 80 2.4. Reflexive Banach Spaces......Page 95 2.5. Problems......Page 116 Chapter 3. The Weak and Weak* Topologies......Page 124 3.1. Topological Vector Spaces......Page 125 3.2. The Banach–Alaoglu Theorem......Page 139 3.3. The Banach–Dieudonné Theorem......Page 145 3.4. The Eberlein–Šmulyan Theorem......Page 149 3.5. The Kreĭn–Milman Theorem......Page 155 3.6. Ergodic Theory......Page 159 3.7. Problems......Page 168 Chapter 4. Fredholm Theory......Page 178 4.1. The Dual Operator......Page 179 4.2. Compact Operators......Page 188 4.3. Fredholm Operators......Page 194 4.4. Composition and Stability......Page 199 4.5. Problems......Page 204 Chapter 5. Spectral Theory......Page 212 5.1. Complex Banach Spaces......Page 213 5.2. Spectrum......Page 223 5.3. Operators on Hilbert Spaces......Page 237 5.4. Functional Calculus for Self-Adjoint Operators......Page 249 5.5. Gelfand Spectrum and Normal Operators......Page 261 5.6. Spectral Measures......Page 276 5.7. Cyclic Vectors......Page 296 5.8. Problems......Page 303 6.1. Unbounded Operators on Banach Spaces......Page 310 6.2. The Dual of an Unbounded Operator......Page 321 6.3. Unbounded Operators on Hilbert Spaces......Page 328 6.4. Functional Calculus and Spectral Measures......Page 341 6.5. Problems......Page 357 Chapter 7. Semigroups of Operators......Page 364 7.1. Strongly Continuous Semigroups......Page 365 7.2. The Hille–Yosida–Phillips Theorem......Page 378 7.3. The Dual Semigroup......Page 392 7.4. Analytic Semigroups......Page 403 7.5. Banach Space Valued Measurable Functions......Page 419 7.6. Inhomogeneous Equations......Page 440 7.7. Problems......Page 454 A.1. The Lemma of Zorn......Page 460 A.2. Tychonoff’s Theorem......Page 464 Bibliography......Page 468 Notation......Page 474 Index......Page 476 Back Cover......Page 482 Functional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, and physics. This book provides a comprehensive introduction to the field for graduate students and researchers. It begins in Chapter 1 with an introduction to the necessary foundations, including the Arzelà–Ascoli theorem, elementary Hilbert space theory, and the Baire Category Theorem. Chapter 2 develops the three fundamental principles of functional analysis (uniform boundedness, open mapping theorem, Hahn–Banach theorem) and discusses reflexive spaces and the James space. Chapter 3 introduces the weak and weak$^•$ topologies and includes the theorems of Banach–Alaoglu, Banach–Dieudonné, Eberlein–Šmulyan, Kreibreven–Milman, as well as an introduction to topological vector spaces and applications to ergodic theory. Chapter 4 is devoted to Fredholm theory. It includes an introduction to the dual operator and to compact operators, and it establishes the closed image theorem. Chapter 5 deals with the spectral theory of bounded linear operators. It introduces complex Banach and Hilbert spaces, the continuous functional calculus for self-adjoint and normal operators, the Gelfand spectrum, spectral measures, cyclic vectors, and the spectral theorem. Chapter 6 introduces unbounded operators and their duals. It establishes the closed image theorem in this setting and extends the functional calculus and spectral measure to unbounded self-adjoint operators on Hilbert spaces. Chapter 7 gives an introduction to strongly continuous semigroups and their infinitesimal generators. It includes foundational results about the dual semigroup and analytic semigroups, an exposition of measurable functions with values in a Banach space, and a discussion of solutions to the inhomogeneous equation and their regularity properties. The appendix establishes the equivalence of the Lemma of Zorn and the Axiom of Choice, and it contains a proof of Tychonoff's theorem. With 10 to 20 elaborate exercises at the end of each chapter, this book can be used as a text for a one-or-two-semester course on functional analysis for beginning graduate students. Prerequisites are first-year analysis and linear algebra, as well as some foundational material from the second-year courses on point set topology, complex analysis in one variable, and measure and integration. Functional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, and physics. This book provides a comprehensive introduction to the field for graduate students and researchers. It begins in Chapter 1 with an introduction to the necessary foundations, including the Arzela-Ascoli theorem, elementary Hilbert space theory, and the Baire Category Theorem. Chapter 2 develops the three fundamental principles of functional analysis (uniform boundedness, open mapping theorem, Hahn-Banach theorem) and discusses reflexive spaces and the James space. Chapter 3 introduces the weak and weak$^*$ topologies and includes the theorems of Banach-Alaoglu, Banach-Dieudonne, Eberlein-Smulyan, Krein-Milman, as well as an introduction to topological vector spaces and applications to ergodic theory. Chapter 4 is devoted to Fredholm theory. It includes an introduction to the dual operator and to compact operators, and it establishes the closed image theorem. Chapter 5 deals with the spectral theory of bounded linear operators. It introduces complex Banach and Hilbert spaces, the continuous functional calculus for self-adjoint and normal operators, the Gelfand spectrum, spectral measures, cyclic vectors, and the spectral theorem. Chapter 6 introduces unbounded operators and their duals. It establishes the closed image theorem in this setting and extends the functional calculus and spectral measure to unbounded self-adjoint operators on Hilbert spaces. Chapter 7 gives an introduction to strongly continuous semigroups and their infinitesimal generators. It includes foundational results about the dual semigroup and analytic semigroups, an exposition of measurable functions with values in a Banach space, and a discussion of solutions to the inhomogeneous equation and their regularity properties. The appendix establishes the equivalence of the Lemma of Zorn and the Axiom of Choice, and it contains a proof of Tychonoff's theorem. With 10 to 20 elaborate exercises at the end of each chapter, this book can be used as a text for a one-or-two-semester course on functional analysis for beginning graduate students. Prerequisites are first-year analysis and linear algebra, as well as some foundational material from the second-year courses on point set topology, complex analysis in one variable, and measure and integration. Introduction -- Foundations -- Principles Of Functional Analysis -- The Weak And Weak* Topologies -- Fredholm Theory -- Spectral Theory -- Unbounded Operators -- Semigroups Of Operators. Theo Bühler, Dietmar A. Salamon. Includes Bibliographical References And Index.