Universitext Gheorghe Moroşanu Functional Analysis for the Applied Sciences Universitext Universitext Series editors SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA CarlesCasacuberta UniversitatdeBarcelona,Barcelona,Spain JohnGreenlees UniversityofWarwick,Coventry,UK AngusMacIntyre QueenMaryUniversityofLondon,London,UK KennethRibet UniversityofCalifornia,Berkeley,CA,USA ClaudeSabbah E´colePolytechnique,CNRS,Universite´ Paris-Saclay,Palaiseau,France EndreSu¨li UniversityofOxford,Oxford,UK WojborA.Woyczyn´ski CaseWesternReserveUniversity,Cleveland,OH,USA Universitext is a series of textbooks that presents material from a wide variety of mathematicaldisciplinesatmaster’slevelandbeyond. Thebooks,oftenwellclass- testedbytheirauthor,mayhaveaninformal,personalevenexperimentalapproach totheirsubjectmatter. Someofthemostsuccessfulandestablishedbooksinthese- rieshaveevolvedthroughseveraleditions,alwaysfollowingtheevolutionofteach- ingcurricula,intoverypolishedtexts. Thus as research topics trickle down into graduate-level teaching, first textbooks writtenfornew,cutting-edgecoursesmaymaketheirwayintoUniversitext. Moreinformationaboutthisseriesathttp://www.springer.com/series/223 Gheorghe Moros¸anu Functional Analysis for the Applied Sciences GheorgheMoros¸anu RomanianAcademyofSciences Bucharest,Romania DepartmentofMathematics Babes-BolyaiUniversity Cluj-Napoca,Romania ISSN0172-5939 ISSN2191-6675 (electronic) Universitext ISBN978-3-030-27152-7 ISBN978-3-030-27153-4 (eBook) https://doi.org/10.1007/978-3-030-27153-4 MathematicsSubjectClassification(2010):32A70 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublica- tiondoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromthe relevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication. Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Dedicated to my wife, Carmen Preface The goal of this book is to present in a friendly manner some of the main results and techniques in Functional Analysis and use them to explore various areas in mathematics and its applications. Special attention is paid to creating appropriate frameworks towards solving different problems in the field of differential and integral equations. In fact, the flavor of this book is given by the fine interplay between the tools offered by Functional Analysis and some specific problems which are of interest in the Applied Sciences. The table of contents of the book (see below) offers a fairly good description of the material. In contrast with other books in the field, we present in Chap.1 the real number system, describing the Cantor– M´eraymodelwhichismostappropriateforourpurposeshere. Indeed, it is based on a completion procedure, allowing the extension from ra- tional numbers to real numbers. This procedure involves the concepts of limit and infinity that are specific to analysis. We consider the Cantor–M´eray construction as the corner stone of mathematical anal- ysis, which is why we pay attention to this subject which is usually assumed well known. In order to help the reader to understand the richness of ideas and methods offered by Functional Analysis, we have included a section of exercises at the end of each chapter. Some of these exercises supple- ment the theoretical material discussed in the corresponding chapter, while others are mathematical problems that are related to the real world. Some of the exercises are borrowed from other books, being reformulated and/or presented in a form adapted to the needs of the correspondingchapter. Wedonotindicatethebookswhereindividual exercises come from, but all those sources are included into the refer- ence list of our book. In any event, we do not claim originality in such cases. Other exercises were invented by us to offer the reader enough vii viii Preface material to understand the theoretical part of the book and gain ex- pertise in solving practical problems. In the last chapter of the book (Chap.12), we provide solutions to almost all exercises. This is incon- trast to many other books which include exercises without solutions. For easy exercises, we provide hints or final solutions, and answers to very easy exercises are left to the reader. I encourage everybody to spend some time working on an exercise before looking at its solution. We shall refer to an exercise by indicating the chapter and exercise numbers (and not the section number). For example, Exercise 11.3 will mean Exercise 3 in the last section of Chap.11 (which is Sect.11.3 in this case). The book is addressed to graduate students and researchers in applied mathematics and neighboring fields of science. I would like to thank the anonymous reviewers whose pertinent comments improved the initial version of the book. Special thanks are due to a former American student of mine, Ivan Andrus, who wrote the first draft of the present book as lecture notes for my Functional Analysis lectures in 2010. He also carefully checked the final version of the book and suggested several minor changes. I am also indebted to my former student Liviu Nicolaescu for read- ing the first part of the book and correcting some errors. Last but not least, I would like to thank Mrs. Elizabeth Loew, Executive Editor at Springer, for our very kind cooperation that led to the successful completion of this book project. Cluj-Napoca, Romania Gheorghe Moro¸sanu Contents 1 Introduction 1 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Complex Numbers . . . . . . . . . . . . . . . . . . . . 15 1.5 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . 16 1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Metric Spaces 31 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Continuous Functions on Compact Sets . . . . . . . . . 44 2.5 The Banach Contraction Principle . . . . . . . . . . . 55 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3 The Lebesgue Integral and Lp Spaces 65 3.1 Measurable Sets in Rk . . . . . . . . . . . . . . . . . . 65 3.2 Measurable Functions . . . . . . . . . . . . . . . . . . . 71 3.3 The Lebesgue Integral . . . . . . . . . . . . . . . . . . 75 3.4 Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4 Continuous Linear Operators and Functionals 89 4.1 Definitions, Examples, Operator Norm . . . . . . . . . 89 4.2 Main Principles of Functional Analysis . . . . . . . . . 93 4.3 Compact Linear Operators . . . . . . . . . . . . . . . . 96 4.4 Linear Functionals, Dual Spaces, Weak Topologies . . 97 4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 104 ix x Contents 5 Distributions, Sobolev Spaces 107 5.1 Test Functions . . . . . . . . . . . . . . . . . . . . . . . 107 5.2 Friedrichs’ Mollification. . . . . . . . . . . . . . . . . . 112 5.3 Scalar Distributions . . . . . . . . . . . . . . . . . . . . 119 5.3.1 Some Operations with Distributions . . . . . . 121 5.3.2 Convergence in Distributions . . . . . . . . . . 122 5.3.3 Differentiation of Distributions . . . . . . . . . 125 5.3.4 Differential Equations for Distributions . . . . . 131 5.4 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . 143 5.5 Bochner’s Integral . . . . . . . . . . . . . . . . . . . . . 149 5.6 Vector Distributions, Wm,p(a,b; X) Spaces . . . . . . . 155 5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6 Hilbert Spaces 165 6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.2 Jordan–von Neumann Characterization Theorem . . . 168 6.3 Projections in Hilbert Spaces . . . . . . . . . . . . . . 171 6.4 The Riesz Representation Theorem . . . . . . . . . . . 175 6.5 Lax–Milgram Theorem . . . . . . . . . . . . . . . . . . 180 6.6 Fourier Series Expansions . . . . . . . . . . . . . . . . 186 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7 Adjoint, Symmetric, and Self-adjoint Linear Operators 201 7.1 The Adjoint of a Linear Operator . . . . . . . . . . . . 201 7.2 Adjoints of Operators on Hilbert Spaces . . . . . . . . 204 7.2.1 The Case of Compact Operators . . . . . . . . 205 7.3 Symmetric Operators and Self-adjoint Operators . . . 209 7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 212 8 Eigenvalues and Eigenvectors 217 8.1 Definition and Examples . . . . . . . . . . . . . . . . . 217 8.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . 219 8.3 Eigenvalues of −Δ Under the Dirichlet Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . 226 8.4 Eigenvalues of −Δ Under the Robin Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.5 Eigenvalues of −Δ Under the Neumann Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . 230 8.6 Some Comments . . . . . . . . . . . . . . . . . . . . . 232 8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 239