Principles of Functional Analysis: Second Edition

Main subject categories: • Functional analysisFunctional analysis plays a crucial role in the applied sciences as well as in mathematics. It is a beautiful subject that can be motivated and studied for its own sake. In keeping with this basic philosophy, the author has made this introductory text accessible to a wide spectrum of students, including beginning-level graduates and advanced undergraduates. The exposition is inviting, following threads of ideas, describing each as fully as possible, before moving on to a new topic. Supporting material is introduced as appropriate, and only to the degree needed. Some topics are treated more than once, according to the different contexts in which they arise.The prerequisites are minimal, requiring little more than advanced calculus and no measure theory. The text focuses on normed vector spaces and their important examples, Banach spaces and Hilbert spaces. The author also includes topics not usually found in texts on the subject. This Second Edition incorporates many new developments while not overshadowing the book's original flavor. Areas in the book that demonstrate its unique character have been strengthened. In particular, new material concerning Fredholm and semi-Fredholm operators is introduced, requiring minimal effort as the necessary machinery was already in place. Several new topics are presented, but relate to only those concepts and methods emanating from other parts of the book. These topics include perturbation classes, measures of noncompactness, strictly singular operators, and operator constants. Overall, the presentation has been refined, clarified, and simplified, and many new problems have been added. Machine generated contents note: Chapter 1. BASIC NOTIONS 1.1. A problem from differential equations 1.2. An examination of the results 1.3. Examples of Banach spaces 1.4. Fourier series 1.5. Problems Chapter 2. DUALITY 2.1. The Riesz representation theorem 2.2. The Hahn-Banach theorem 2.3. Consequences of the Hahn-Banach theorem 2.4. Examples of dual spaces 2.5. Problems Chapter 3. LINEAR OPERATORS 3.1. Basic properties 3.2. The adjoint operator 3.3. Annihilators 3.4. The inverse operator 3.5. Operators with closed ranges 3.6. The uniform boundedness principle 3.7. The open mapping theorem 3.8. Problems Chapter 4. THE RIESZ THEORY FOR COMPACT OPERATORS 4.1. A type of integral equation 4.2. Operators of finite rank 4.3. Compact operators 4.4. The adjoint of a compact operator 4.5. Problems Chapter 5. FREDHOLM OPERATORS 5.1. Orientation 5.2. Further properties 5.3. Perturbation theory 5.4. The adjoint operator 5.5. A special case 5.6. Semi-Fredholm operators 5.7. Products of operators 5.8. Problems Chapter 6. SPECTRAL THEORY 6.1. The spectrum and resolvent sets 6.2. The spectral mapping theorem 6.3. Operational calculus 6.4. Spectral projections 6.5. Complexification 6.6. The complex Hahn-Banach theorem 6.7. A geometric lemma 6.8. Problems Chapter 7. UNBOUNDED OPERATORS 7.1. Unbounded Fredholm operators 7.2. Further properties 7.3. Operators with closed ranges 7.4. Total subsets 7.5. The essential spectrum 7.6. Unbounded semi-Fredholm operators 7.7. The adjoint of a product of operators 7.8. Problems Chapter 8. REFLEXIVE BANACH SPACES 8.1. Properties of reflexive spaces 8.2. Saturated subspaces 8.3. Separable spaces 8.4. Weak convergence 8.5. Examples 8.6. Completing a normed vector space 8.7. Problems Chapter 9. BANACH ALGEBRAS 9.1. Introduction 9.2. An example 9.3. Commutative algebras 9.4. Properties of maximal ideals 9.5. Partially ordered sets 9.6. Riesz operators 9.7. Fredholm perturbations 9.8. Semi-Fredholm perturbations 9.9. Remarks 9.10. Problems Chapter 10. SEMIGROUPS 10.1. A differential equation 10.2. Uniqueness 10.3. Unbounded operators 10.4. The infinitesimal generator 10.5. An approximation theorem 10.6. Problems Chapter 11. HILBERT SPACE 11.1. When is a Banach space a Hilbert space? 11.2. -Normal operators 11.3. Approximation by operators of finite rank 11.4. Integral operators 11.5. Hyponormal operators 11.6. Problems Chapter 12. BILINEAR FORMS 12.1. The numerical range 12.2. The associated operator 12.3. Symmetric forms 12.4. Closed forms 12.5. Closed extensions 12.6. Closable operators 12.7. Some proofs 12.8. Some representation theorems 12.9. Dissipative operators 12.10. The case of a line or a strip 12.11. Selfadjoint extensions 12.12. Problems Chapter 13. SELFADJOINT OPERATORS 13.1. Orthogonal projections 13.2. Square roots of operators 13.3. A decomposition of operators 13.4. Spectral resolution 13.5. Some consequences 13.6. Unbounded selfadjoint operators 13.7. Problems Chapter 14. MEASURES OF OPERATORS 14.1. A seminorm 14.2. Perturbation classes 14.3. Related measures 14.4. Measures of noncompactness 14.5. The quotient space 14.6. Strictly singular operators 14.7. - Norm perturbations 14.8. Perturbation functions 14.9. Factored perturbation functions 14.10. Problems Chapter 15. EXAMPLES AND APPLICATIONS 15.1. A few remarks 15.2. A differential operator 15.3. Does A have a closed extension? 15.4. The closure of A 15.5. Another approach 15.6. The Fourier transform 15.7. Multiplication by a function 15.8. More general operators 15.9. B-Compactness 15.10. The adjoint of A 15.11. An integral operator 15.12. Problems Appendix A. Glossary Appendix B. Major Theorems Bibliography Index. This excellent book provides an elegant introduction to functional analysis ... carefully selected problems ... This is a nicely written book of great value for stimulating active work by students. It can be strongly recommended as an undergraduate or graduate text, or as a comprehensive book for self-study. —European Mathematical Society Newsletter Functional analysis plays a crucial role in the applied sciences as well as in mathematics. It is a beautiful subject that can be motivated and studied for its own sake. In keeping with this basic philosophy, the author has made this introductory text accessible to a wide spectrum of students, including beginning-level graduates and advanced undergraduates. The exposition is inviting, following threads of ideas, describing each as fully as possible, before moving on to a new topic. Supporting material is introduced as appropriate, and only to the degree needed. Some topics are treated more than once, according to the different contexts in which they arise. The prerequisites are minimal, requiring little more than advanced calculus and no measure theory. The text focuses on normed vector spaces and their important examples, Banach spaces and Hilbert spaces. The author also includes topics not usually found in texts on the subject. This Second Edition incorporates many new developments while not overshadowing the book's original flavor. Areas in the book that demonstrate its unique character have been strengthened. In particular, new material concerning Fredholm and semi-Fredholm operators is introduced, requiring minimal effort as the necessary machinery was already in place. Several new topics are presented, but relate to only those concepts and methods emanating from other parts of the book. These topics include perturbation classes, measures of noncompactness, strictly singular operators, and operator constants. Overall, the presentation has been refined, clarified, and simplified, and many new problems have been added. The book is recommended to advanced undergraduates, graduate students, and pure and applied research mathematicians interested in functional analysis and operator theory. Cover 1 Other titles in this series 2 Title page 6 Contents 10 Preface to the revised edition 16 From the preface to the first edition 20 Basic notions 24 Duality 52 Linear operators 78 The Riesz theory for compact operators 100 Fredholm operators 124 Spectral theory 152 Unbounded operators 178 Reflexive Banach spaces 206 Banach algebras 224 Semigroups 248 Hilbert space 266 Bilinear forms 288 Selfadjoint operators 320 Measures of operators 348 Examples and applications 382 Glossary 416 Major Theorems 428 Bibliography 442 Index 446 Back Cover 450 Functional analysis plays a crucial role in the applied sciences as well as in mathematics. In keeping with this fact the author has made this advanced textbook accessible to a wide range of students including advanced undergraduates Focuses on normed vector spaces and their important examples, Banach spaces and Hilbert spaces. This work covers topics including perturbation classes, measures of noncompactness, strictly singular operators and operator constants.