Problems in Real and Functional Analysis (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 166)

It is generally believed that solving problems is the most important part of the learning process in mathematics because it forces students to truly understand the definitions, comb through the theorems and proofs, and think at length about the mathematics. The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of conceptual problems (1,457 in total), ranging from easily accessible to thought provoking, mixing the practical and the theoretical aspects of the subject. Problems are grouped into ten chapters covering the main topics usually taught in courses on real and functional analysis. Each of these chapters opens with a brief reader's guide stating the needed definitions and basic results in the area and closes with a short description of the problems. The Problem chapters are accompanied by Solution chapters, which include solutions to two-thirds of the problems. Students can expect the solutions to be written in a direct language that they can understand; usually the most "natural"; rather than the most elegant solution is presented. Readership Graduate students and researchers interested in learning and teaching real and functional analysis at the graduate level. Part I. Problems -- Chapter I. Set Theory And Metrie Spaees, P.3 -- Problems, P.6 -- Chapter 2 Measures, P.13 -- Problems, P.15 -- Chapter 3. Lebesgue Measure, P.29 -- Problems, P.30 -- Chapter 4. Measurable And Integrable Functions, P.41 -- Problems, P.44 -- Chapter 5. Lp Spaces, P.59 -- Problems, P.60 -- Chapter 6. Sequenees Of Funetions, P.75 -- Problems, P.76 -- Chapter 7. Produet Measures, P.93 -- Problems, P.95 -- Chapter 8. Normed Linear Spaces Functionals, P.105 -- Problems, P.108 -- Chapter 9. Normed Linear Spaces. Linear Operators, P.125 -- Problems, P.127 -- Chapter 10. Hilbert Spaces, P.147 -- Problems, P.150 -- Part 2. Solutions -- Chapter 11. Set Theory And Metrie Spaces, P.169 -- Solutions, P.169 -- Chapter 12. Measures, P.191 -- Solutions, P.191 -- Chapter 13. Lebesgue Measure, P.221 -- Solutions, P.221 -- Chapter 14. Measurable And Integrable Functions, P.249 -- Solutions, P.249 -- Chapter 15. Lp Spaees, P.283 -- Solutions, P.283 -- Chapter 16. Sequenees Of Functions, P.315 -- Solutions, P.315 -- Chapter 17. Product Measures, P.349 -- Solutions, P.349 -- Chapter 18. Solutions, P.365 -- Chapter 19. Normed Linear Spaces. Linear Operators, P.403 -- Solutions, P.403 -- Chapter 20. Hilbert Spaces, P.433 -- Solutions, P.433 -- Index, P.465. Alberto Torchinsky. Includes Index. Includes Bibliographical References And Index. Preface ix Part 1. Problems Chapter 1. Set Theory and Metric Spaces 3 Problems 6 Chapter 2. Measures 13 Problems 15 Chapter 3. Lebesgue Measure 29 Problems 30 Chapter 4. Measurable and Integrable Functions 41 Problems 44 Chapter 5. Lp Spaces 59 Problems 60 Chapter 6. Sequences of Functions 75 Problems 76 Chapter 7. Product Measures 93 Problems 95 Chapter 8. Normed Linear Spaces. Functionals 105 Problems 108 Chapter 9. Normed Linear Spaces. Linear Operators 125 Problems 127 Chapter 10. Hilbert Spaces 147 Problems 150 Part 2. Solutions Chapter 11. Set Theory and Metric Spaces 169 Solutions 169 Chapter 12. Measures 191 Solutions 191 Chapter 13. Lebesgue Measure 221 Solutions 221 Chapter 14. Measurable and Integrable Functions 249 Solutions 249 Chapter 15. Lp Spaces 283 Solutions 283 Chapter 16. Sequences of Functions 315 Solutions 315 Chapter 17. Product Measures 349 Solutions 349 Chapter 18. Normed Linear Spaces. Functionals 365 Solutions 365 Chapter 19. Normed Linear Spaces. Linear Operators 403 Solutions 403 Chapter 20. Hilbert Spaces 433 Solutions 433 Index 465 Cover -- Title Page -- Dedication -- Contents -- Preface -- Part 1. Problems -- Chapter 1. Set Theory And Metric Spaces -- Chapter 2. Measures -- Chapter 3. Lebesgue Measure -- Chapter 4. Measurable And Integrable Functions -- Chapter 5. ^{ } Spaces -- Chapter 6. Sequences Of Functions -- Chapter 7. Product Measures -- Chapter 8. Normed Linear Spaces. Functionals -- Chapter 9. Normed Linear Spaces. Linear Operators -- Chapter 10. Hilbert Spaces -- Part 2. Solutions -- Chapter 11. Set Theory And Metric Spaces -- Chapter 12. Measures -- Chapter 13. Lebesgue Measure -- Chapter 14. Measurable And Integrable Functions -- Chapter 15. ^{ } Spaces -- Chapter 16. Sequences Of Functions -- Chapter 17. Product Measures -- Chapter 18. Normed Linear Spaces. Functionals -- Chapter 19. Normed Linear Spaces. Linear Operators -- Chapter 20. Hilbert Spaces -- Index -- Back Cover "The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of conceptual problems (1,457 in total), ranging from easily accessible to thought provoking, mixing the practical and the theoretical aspects of the subject. Problems are grouped into ten chapters covering the main topics usually taught in courses on real and functional analysis."--Page 4 de la couverture