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Rajnikant Sinha Real and Complex Analysis Volume 1 Real and Complex Analysis Rajnikant Sinha Real and Complex Analysis Volume 1 123 Rajnikant Sinha Varanasi, Uttar Pradesh, India ISBN978-981-13-0937-3 ISBN978-981-13-0938-0 (eBook) https://doi.org/10.1007/978-981-13-0938-0 LibraryofCongressControlNumber:2018945438 ©SpringerNatureSingaporePteLtd.2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721, Singapore Preface The book is an introduction to real and complex analysis that will be useful to undergraduate students of mathematics and engineering. It is designed to equip the readerwithtools thatwill help themtounderstandthe conceptsofrealanalysis and complex analysis. In addition, it contains the essential topics of analysis that are neededforthestudyoffunctionalanalysis.Itsguidingprincipleistohelpdevelopthe necessary concepts rigorously with enough detail and with the minimum prerequi- sites.Further,Ihaveendeavoredtomakethisbookbothaccessibleandreadable.This book contains complete solutions to almost all the problems discussed within. This willbebeneficialtoreadersonlyifusedcorrectly:readersareencouragedtolookat the solution to a problem only after trying to solve the problem. Certainly,attimes,thereadermayfindtheproofsexcruciatinglydetailed,butit is better to be detailed than concise. Furthermore, omitting the detailed calculation can sometimes be perplexing for beginners. I have tried to make it a readable text that caters to a broad audience. This approach should certainly benefit beginners who have not yet tussled with the subject in a serious way. This book contains several useful theorems and their proofs in the realm of real and complex analysis. Most of these theorems are the works of some the great mathematicians of the 19th and 20th centuries. In alphabetical order, some of these include: Arzela, Ascoli, Baire, Banach, Carathéodory, Cauchy, Dirichlet, Egoroff, Fatou, Fourier, Fubini, Hadamard, Jordan, Lebesgue, Liouville, Minkowski, Mittag-Leffler, Morera, Nikodym, Ostrowski, Parseval, Picard, Plancherel, Poisson, Radon,Riemann,Riesz,Runge,Schwarz,Taylor,Tietze,Urysohn,Weierstrass,and Young. I have spent several years providing their proofs in unprecedented detail. There are plenty of superb texts on real and complex analysis, but there is a dearth of books that blend real analysis with complex analysis. Libraries already contain several excellent reference books on real and complex analysis, which interested students can consult for a deeper understanding. It was notmy intention to replace such books. This book is written under the assumption that students alreadyknowthefundamentalsofadvancedcalculus.Theproofsofvariousnamed v vi Preface theorems should be considered to be at the core of the book by any reader who is serious about learning the subject. The book is divided into two volumes. Volume 1 contains three chapters: Lebesgueintegration,Lp-spacesandFouriertransforms.InChap.1,webeginwith the definition of an exponential function and prove that it maps the set of all complex numbers onto the set of all nonzero complex numbers. After that, we develop the Lebesgue theory of abstract integration of complex-valued functions. Next, we prove the Riesz representation theorem in enough detail and use it to answerthe question: is every set of n-tuples of real numbersLebesgue measurable in Rn? The theme of Chap. 2 is Lp-spaces. First of all, we introduce convex functions and then prove the Riesz–Fischer theorem. In the end, we derive some properties of Banach algebra. We have shown that Lp-spaces is an example of a Banach space. In Chap. 3, we introduce total variation, and prove the Radon– Nikodym theorem. Fubini theorem and the change-of-variable theorem are also proved in this chapter. Finally, we discuss the Plancherel theorem on Fourier transforms. I am particularly indebted to Walter Rudin and Paul Richard Halmos for their letters discussing academic questions. By great good fortune, some colleagues of minewereabletojoininwiththisenterpriseafewyearsago,someofwhomhave provideda meticulous reading of themanuscript from a user’sviewpoint. Iextend my great thanks to all of them for their expert services. Whilestudyingthisbook,Ihopethatreaderswillexperiencethethrillofcreative effort and the joy of achievement. Varanasi, India Rajnikant Sinha Contents 1 Lebesgue Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3 Integration of Positive Functions . . . . . . . . . . . . . . . . . . . . . . . . 51 1.4 Integration of Complex-Valued Functions . . . . . . . . . . . . . . . . . 85 1.5 Sets of Measure Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 1.6 Preliminaries to Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 1.7 Preliminaries to Riesz Representation Theorem. . . . . . . . . . . . . . 139 1.8 Riesz Representation Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 167 1.9 Borel Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 1.10 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 1.11 Existence of Non-Lebesgue Measurable Sets . . . . . . . . . . . . . . . 216 2 Lp-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 2.1 Convex Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 2.2 The Lp-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 2.3 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 2.4 Orthogonal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 2.5 Riesz-Fischer Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 2.6 Baire’s Category Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 2.7 Hahn-Banach Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 2.8 Banach Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 3 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 3.1 Total Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 3.2 Radon–Nikodym Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 3.3 Bounded Linear Functionals on Lp. . . . . . . . . . . . . . . . . . . . . . . 431 3.4 Lebesgue Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 3.5 Metric Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 3.6 Vitali–Caratheodory Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 510 vii viii Contents 3.7 Change-of-Variables Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 535 3.8 Fubini Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 3.9 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 3.10 Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 3.11 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 3.12 Inversion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 3.13 Plancherel Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 Bibliography .. .... .... .... ..... .... .... .... .... .... ..... .... 637 About the Author RajnikantSinha isFormerProfessorofMathematicsatMagadhUniversity,Bodh Gaya,India.Asapassionatemathematician,hehaspublishednumerousinteresting research findings in international journals and books, including Smooth Manifolds (Springer)andthecontributedbookSolutionstoWeatherburn’sElementaryVector Analysis. His research focuses on topological vector spaces, differential geometry and manifolds. ix Chapter 1 Lebesgue Integration InadequaciesoftheRiemannintegration,anddifficultiesinhandlinglimitprocesses in it were largely overcome with the advent of abstract integration. Next, upon applying a remarkable result—the Riesz representation theorem—Lebesgue measure in Euclidean space is introduced. Just like the real number system is a completion,inacertainsense,oftherationalnumbersystem,Lebesgueintegrationisa completion, in a certain sense, of Riemann integration. These phenomena will be vividly demonstrated in this somewhat long chapter. We begin with the definition ofexponentialfunction,andprovethatitmapsthesetofallcomplexnumbersonto thesetofallnonzerocomplexnumbers.Afterthat,wedeveloptheLebesguetheory of abstract integration of complex-valued functions. Next we prove the Riesz representationtheoreminsufficientdetailanduseittoanswerthequestion:Isevery set of n-tuples of real numbers is Lebesgue measurable in Rn? 1.1 Exponential Function Since exponential function will occur quite frequently in later chapters, it seems prudenttolayagoodfoundationforthisattheearliestopportunity.Thissectionis devoted to this end. Note 1.1 For every complex number z, (cid:1) (cid:1) lim(cid:1)(cid:1)(cid:1)ðnznþþ11Þ!(cid:1)(cid:1)(cid:1)¼ lim jzj ¼jzj(cid:3)lim 1 (cid:4)¼jzj(cid:2)0¼0\1; n!1(cid:1) znn! (cid:1) n!1nþ1 n!1nþ1 so, by the ratio test of convergence, the series 1þzþ z2 þ z3 þ (cid:2)(cid:2)(cid:2) is absolutely 2! 3! convergent for every complex number z. ©SpringerNatureSingaporePteLtd.2018 1 R.Sinha,RealandComplexAnalysis, https://doi.org/10.1007/978-981-13-0938-0_1