Alberto Torchinsky Problems in Real and Functional Analysis Problems in Real and Functional Analysis Alberto Torchinsky Graduate Studies in Mathematics Volume 166 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 26-01, 28-01, 46-01, 47-01. For additional information and updates on this book, visit www. ams. org/bookpages/gsm-166 Library of Congress Cataloging-in-Publication Data Torchinsky, Alberto. Problems in real and functional analysis / Alberto Torchinsky. pages cm. — (Graduate studies in mathematics ; volume 166) Includes index. ISBN 978-1-4704-2057-4 (alk. paper) 1. Mathematical analysis—Textbooks. 2. Functional analysis—Textbooks. 3. Set theory— Textbooks. I. Title. QA300.T65 2015 515,.7-dc23 2015022653 Copying and reprinting. 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Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. © 2015 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 20 19 18 17 16 15 To Massi Contents 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. IP 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 Vil vm Contents 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. IP 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 Preface Students tell me that they learn mathematics primarily from doing prob­ lems. They say that a good course is one that motivates the material dis­ cussed, building on basic concepts and ideas leading to abstract generality, one that presents the “big picture” rather than isolated theorems and results. And, they say that problems are the most important part of the learning process, because the problems force them to truly understand the defini­ tions, comb through the proofs and theorems, and think at length about the mathematics. Exercises that require basic application of the theorems highlight the power of the theorems. They also offer an opportunity to encourage stu­ dents to construct examples for themselves. Problems can also be used to explore counterexamples to conjectures. Supplying a counterexample helps the student gain insight into theorems, including an understanding of the necessity of the assumptions. Well-crafted problems review and expand on the material and give students a chance to participate in the mathematical process. Open-ended problems (“Discuss the validity of... ”) afford the stu­ dents the opportunity to adjust to researching and discovering mathematics for themselves. The purpose of this book is to complement the existing literature in in­ troductory real and functional analysis at the graduate level with a variety of conceptual problems, ranging from readily accessible to thought provoking, mixing the practical and the theoretical. Students can expect the solutions to be written in a direct language, one they can understand; always the most “natural” rather than the most elegant solution is presented. The book consists of twenty chapters: Chapters 1 through 10 contain the Problems, and Chapters 11 to 20 contain (selected) Solutions. Chapters IX