A Complete Solution Guide to Real and Complex Analysis I by Kit-Wing Yu, PhD [email protected] Copyright c 2019 by Kit-Wing Yu. All rights reserved. No part of this publication may be (cid:13) reproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the author. ISBN: 978-988-78797-8-7 (eBook) ISBN: 978-988-78797-9-4 (Paperback) ii About the author Dr. Kit-Wing Yu received his B.Sc. (1st Hons), M.Phil. and Ph.D. degrees in Math. at the HKUST,PGDE (Mathematics) at the CUHK.After his graduation, hehas joined United Chris- tian College to serve as a mathematics teacher for at least seventeen years. He has also taken the responsibility of the mathematics panel since 2002. Furthermore, he was appointed as a part-time tutor (2002 – 2005) and then a part-time course coordinator (2006 – 2010) of the Department of Mathematics at the OUHK. Besides teaching, Dr. Yu has been appointed to bea marker of the HKAL PureMathematics and HKDSE Mathematics (Core Part) for over thirteen years. Between 2012 and 2014, Dr. Yu was invited tobeaJudgeMemberbytheWorld OlympicMathematics Competition(China). In the research aspect, he has published over twelve research papers in international mathematical journals, including some well-known journals such as J. Reine Angew. Math., Proc. Roy. Soc. EdinburghSect. Aand Kodai Math. J.. His research interests are inequalities, special functions and Nevanlinna’s value distribution theory. iii iv Preface ProfessorWalter Rudina istheauthoroftheclassical andfamoustextbooks: Principles of Math- ematical Analysis, Real and Complex Analysis, and Functional Analysis. (People commonly call them “Baby Rudin”, “Papa Rudin” and “Grandpa Rudin” respectively.) Undoubtedly, they have produced important and extensive impacts to the study of mathematical analysis at uni- versity level since their publications. As far as you know, Real and Complex Analysis keeps the features of Principles of Math- ematical Analysis which are well-organized and expositions of theorems are clear, precise and well-written. Therefore, Real and Complex Analysis is always one of the main textbooks or references of graduate real analysis course in many universities. Actually, some universities will request their Ph.D. students to study this book for their qualifying examinations. After the publication of the book A Complete Solution Guide to Principles of Mathematical Analysis,somepurchasershavesuggestedmetowriteasolutionbookofReal and Complex Anal- ysis. (To the best of the author’s knowledge, Papa Rudin has no solution manual.) I was afraid of doing so at the beginning because the exercises are at graduate level and they are much more difficult than those in Baby Rudin. Fortunately, I was used to keeping solutions of mathematics exercises done by me in my undergraduate and graduate study. In fact, I have kept at least 25% of the exercises in the first six chapters of Papa Rudin. Therefore, after thorough considera- tion,IdecidedtostartthenextprojectofasolutionbooktoRudin’sReal and Complex Analysis. Since Papa Rudin consists of two components, I plan to write the solutions for “Real Anal- ysis” part first. In fact, the present book A Complete Solution Guide to Real and Complex Analysis I covers all the exercises of Chapters 1 to 9 and its primary aim is to help every mathematics student and instructor to understand the ideas and applications of the theorems in Rudin’s book. To accomplish this goal, I have adopted the way I wrote the book A Complete Solution Guide to Principles of Mathematical Analysis. In other words, I intend writing the solutions as comprehensive as I can so that you can understand every detailed part of a proof easily. Apart from this, I also keep reminding you what theorems or results I have applied by quoting them repeatedly in the proofs. By doing this, I believe that you will become fully aware of the meaning and applications of each theorem. Before you read this book, I have two gentle reminders for you. Firstly, as a mathematics instructor at a college, I understand that the growth of a mathematics student depends largely on how hard he/she does exercises. When your instructor asks you to do some exercises from Rudin,you arenot suggested to read my solutions unless you have tried your bestto prove them yourselves. Secondly, when I prepared this book, I found that some exercises require knowledge that Rudin did not cover in his book. To fill this gap, I refer to some other analysis or topology ahttps://en.wikipedia.org/wiki/Walter_Rudin. v vi books such as [9], [10], [22], [42] and [47]. Other useful references are [1], [12], [24], [28], [29], [59], [64], [66] and [67]. Of course, it is not a surprise that we will regard the exercises in Baby Rudin as some known facts and if you want to read proofs of them, you are strongly advised to read my book [63]. The features of this book are as follows: It covers all the 176 exercises from Chapters 1 to 9 with detailed and complete solutions. • As a matter of fact, my solutions show every detail, every step and every theorem that I applied. There are 11 illustrations for explaining the mathematical concepts or ideas used behind • the questions or theorems. Sections in each chapter are added so as to increase the readability of the exercises. • Different colors are used frequently in order to highlight or explain problems, lemmas, • remarks, main points/formulas involved, or show the steps of manipulation in some com- plicated proofs. (ebook only) Necessary lemmas with proofs are provided because some questions require additional • mathematical concepts which are not covered by Rudin. Many usefulor relevant references are provided to some questions for your futureresearch. • Since the solutions are written solely by me, you may find typos or mistakes. If you really find such a mistake, please send your valuable comments or opinions to [email protected]. Then I will post the updated errata on my website https://sites.google.com/view/yukitwing/ irregularly. Kit Wing Yu April 2019 List of Figures 2.1 The graph of g on [ 1,1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 n − 2.2 The graphs of g on [0,1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 n,k 2.3 The pictures of V,E ,E V and (E V)c. . . . . . . . . . . . . . . . . . . . . . 57 ′ ′ ′ ∩ ∩ 2.4 The set (E V)c U. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ′ ∩ \ 3.1 The distribution of x and ǫ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 n 3.2 The geometric interpretation of a special case. . . . . . . . . . . . . . . . . . . . . 89 5.1 The unit circle in different p-norm. . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.2 The square K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.1 The graphs of g (x) and g (x). . . . . . . . . . . . . . . . . . . . . . . . . . . 194 n,1 n,2 7.1 The closed intervals E and E . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 n n+1 7.2 Construction of the sequence E (p). . . . . . . . . . . . . . . . . . . . . . . . . . 242 i vii List of Figures viii Contents Preface v List of Figures vii 1 Abstract Integration 1 1.1 Problems on σ-algebras and Measurable Functions . . . . . . . . . . . . . . . . . 1 1.2 Problems related to the Lebesgue’s MCT/DCT . . . . . . . . . . . . . . . . . . . 7 2 Positive Borel Measures 17 2.1 Properties of Semicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Problems on the Lebesgue Measure on R . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Integration of Sequences of Continuous Functions . . . . . . . . . . . . . . . . . . 30 2.4 Problems on Borel Measures and Lebesgue Measures . . . . . . . . . . . . . . . . 36 2.5 Problems on Regularity of Borel Measures . . . . . . . . . . . . . . . . . . . . . . 43 2.6 Miscellaneous Problems on L1 and Other Properties . . . . . . . . . . . . . . . . 59 3 Lp-Spaces 69 3.1 Properties of Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Relations among Lp-Spaces and some Consequences . . . . . . . . . . . . . . . . 71 3.3 Applications of Theorems 3.3, 3.5, 3.8, 3.9 and 3.12 . . . . . . . . . . . . . . . . . 87 3.4 Hardy’s Inequality and Egoroff’s Theorem . . . . . . . . . . . . . . . . . . . . . . 91 3.5 Convergence in Measure and the Essential Range of f L (µ) . . . . . . . . . . 106 ∞ ∈ 3.6 A Converse of Jensen’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.7 The Completeness/Completion of a Metric Space . . . . . . . . . . . . . . . . . . 112 3.8 Miscellaneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4 Elementary Hilbert Space Theory 123 4.1 Basic Properties of Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.2 Application of Theorem 4.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.3 Miscellaneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5 Examples of Banach Space Techniques 143 5.1 The Unit Ball in a Normed Linear Space . . . . . . . . . . . . . . . . . . . . . . . 143 5.2 Failure of Theorem 4.10 and Norm-preserving Extensions . . . . . . . . . . . . . 146 5.3 The Dual Space of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 ix Contents x 5.4 Applications of Baire’s and other Theorems . . . . . . . . . . . . . . . . . . . . . 157 5.5 Miscellaneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6 Complex Measures 183 6.1 Properties of Complex Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.2 Dual Spaces of Lp(µ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.3 Fourier Coefficients of Complex Borel Measures . . . . . . . . . . . . . . . . . . . 191 6.4 Problems on Uniformly Integrable Sets . . . . . . . . . . . . . . . . . . . . . . . . 196 6.5 Dual Spaces of Lp(µ) Revisit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 7 Differentiation 207 7.1 Lebesgue Points and Metric Densities . . . . . . . . . . . . . . . . . . . . . . . . 207 7.2 Periods of Functions and Lebesgue Measurable Groups . . . . . . . . . . . . . . . 210 7.3 The Cantor Function and the Non-measurability of f T . . . . . . . . . . . . . 216 ◦ 7.4 Problems related to the AC of a Function . . . . . . . . . . . . . . . . . . . . . . 218 7.5 Miscellaneous Problems on Differentiation . . . . . . . . . . . . . . . . . . . . . . 232 8 Integration on Product Spaces 249 8.1 Monotone Classes and Ordinate Sets of Functions . . . . . . . . . . . . . . . . . . 249 8.2 Applications of the Fubini Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.3 The Product Measure Theorem and Sections of a Function . . . . . . . . . . . . 268 8.4 Miscellaneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 9 Fourier Transforms 281 9.1 Properties of The Fourier Transforms. . . . . . . . . . . . . . . . . . . . . . . . . 281 9.2 The Poisson Summation Formula and its Applications . . . . . . . . . . . . . . . 297 9.3 Fourier Transforms on Rk and its Applications . . . . . . . . . . . . . . . . . . . 301 9.4 Miscellaneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 Index 315 Bibliography 317