Joseph Muscat Functional Analysis An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras Functional Analysis Joseph Muscat Functional Analysis An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras 123 Joseph Muscat Department of Mathematics Universityof Malta Msida Malta Whilst we have made considerable efforts to contact all holders of copyright material containedinthisbookwehavefailedtolocatesomeofthem.Shouldholderswishtocontact thePublisher,wewill make everyeffort tocome tosomearrangement withthem. ISBN 978-3-319-06727-8 ISBN 978-3-319-06728-5 (eBook) DOI 10.1007/978-3-319-06728-5 Springer ChamHeidelberg New YorkDordrecht London LibraryofCongressControlNumber:2014939386 MathematicsSubjectClassification:46-01,47-01 (cid:2)SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Jennifer-Claire and Bernadette Preface Originally, functional analysis was the study offunctions. It is now considered to be a unifying subject that generalizes much of linear algebra and real/complex analysis, with emphasis on infinite dimensional spaces. This book introduces this vast topic from these elementary preliminaries and develops both the abstract theoryanditsapplicationsinthreeparts:(I)MetricSpaces,(II)BanachandHilbert Spaces, and (III) Banach Algebras. Especially with the digital revolution at the turn of the millennium, Hilbert spaces and least squares approximation have become necessary and fundamental topicsforamathematicaleducation,notonlyjustformathematicians,butalsofor engineers, physicists, and statisticians interested in signal processing, data analy- sis, regression, quantum mechanics, etc. Banach spaces, in particular L1 and L? methods, have gained popularity in applications and are complementing or even supplanting the classical least squares approach to many optimization problems. Aim of this Book The main aim of this book is to provide the reader with an introductory textbook thatstartsfromelementarylinearalgebraandrealanalysisanddevelopsthetheory sufficiently to understand how various applications, including least squares approximation, etc., are all part of a single framework. A textbook must try to achieve a balance between rigor and understanding: not being too elementary by omitting‘hard’proofs,butneithertooadvancedbyusingtoostrictalanguagefor the average reader and treating theorems as mere stepping stones to yet other theorems.Despitethemultitudeofbooksinthisarea,thereisstillaperceivedgap in learning difficulty between undergraduate and graduate textbooks. This book aims to be in the middle: it covers much material and has many exercises of varying difficulty, yet the emphasis is for the student to remember the theory clearly using intuitive language. For example, real analysis is redeveloped from the broader picture of metric spaces (including a construction of the real number space), rather than through the even more abstract topological spaces. vii viii Preface Audience This book is meant for the undergraduate who is interested in mathematical analysisanditsapplications,ortheresearchengineer/statisticianwhowouldlikea more rigorous approach to fundamental mathematical concepts and techniques. It can also serve as a reference or for self-study of a subject that occupies a central place in modern mathematics, opening up many avenues for further study. The basic requirements are mainly the introductory topics of mathematics: Set andLogicnotation,VectorSpaces,andRealAnalysis(calculus).Apartfromthese, itwouldbehelpful,butnotnecessary,tohavetakenelementarycoursesinFourier Series, Lebesgue Integration, and Complex Analysis. Reviews of Vector Spaces and Measurable sets are included in this book, while the other two mentioned subjects are developed only to the extent needed. Examples are included from many areas of mathematics that touch upon functionalanalysis.Itwouldbehelpfulattheappropriateplaces,forthereaderto haveencounteredtheseothersubjects,butthisisnotessential.Theaimistomake connectionsanddescribethemfromtheviewpointoffunctionalanalysis.Withthe modernfacilitiesofsearchingovertheInternet,anyoneinterestedinfollowingup a specific topic can easily do so. The sections follow each other in a linear fashion, with the three parts fitting intothreeone-semestercourses,althoughPartIIistwiceaslongastheothers.The following sections may be omitted without much effect on subsequent topics: Section 6.4 C(X, Y) Section 9.2 Function Spaces Sections 11.5 Pointwise and Weak Convergence Sections 12.1 and 12.2 Differentiation and Integration Sections 14.4 and 14.5 Functional Calculus and the Gelfand Transform Section 15.4 Representation Theorems. Acknowledgments This book grew out of lecture notes that were used in a course on functional analysis for the last 20 years. I wish to thank my students over the years for their feedback, but mostly my colleagues, Emanuel Chetcuti and Josef Lauri for their encouragementandveryhelpfulsuggestionswhenreadingthefirstdraft,aswellas theanonymousreviewerswho,withtheircomments,helpedshapethebook.Iwish tothankalsotheSpringereditors,JoergSixtandCatherineWaite,fortheircarein preparing this manuscript. Malta Joseph Muscat Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Part I Metric Spaces 2 Distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Balls and Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Convergence and Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4 Completeness and Separability . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1 Completeness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Uniformly Continuous Maps . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Separable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1 Connected Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.1 Bounded Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2 Totally Bounded Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.3 Compact Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.4 The Space C(X,Y). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 ix x Contents Part II Banach and Hilbert Spaces 7 Normed Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.2 Norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.3 Metric and Vector Properties. . . . . . . . . . . . . . . . . . . . . . . . 101 7.4 Complete and Separable Normed Vector Spaces . . . . . . . . . . 105 7.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8 Continuous Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.1 Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.2 Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.3 RN and Totally Bounded Sets . . . . . . . . . . . . . . . . . . . . . . . 135 9 Main Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.1 Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.2 Function Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 10 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 10.1 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 10.2 Least Squares Approximation. . . . . . . . . . . . . . . . . . . . . . . . 179 10.3 Duality H(cid:2) (cid:3)H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 10.4 The Adjoint Map T(cid:2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 10.5 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 10.6 Orthonormal Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 11 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 11.1 The Open Mapping Theorem. . . . . . . . . . . . . . . . . . . . . . . . 221 11.2 Compact Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 11.3 The Dual Space X(cid:2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 11.4 The Adjoint T> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 11.5 Pointwise and Weak Convergence . . . . . . . . . . . . . . . . . . . . 246 12 Differentiation and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 257 12.1 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 12.2 Integration for Vector-Valued Functions . . . . . . . . . . . . . . . . 261 12.3 Complex Differentiation and Integration . . . . . . . . . . . . . . . . 267 Part III Banach Algebras 13 Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 13.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 13.2 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Contents xi 13.3 The Group of Invertible Elements . . . . . . . . . . . . . . . . . . . . 295 13.4 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 14 Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 14.1 The Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 14.2 The Spectrum of an Operator. . . . . . . . . . . . . . . . . . . . . . . . 312 14.3 Spectra of Compact Operators . . . . . . . . . . . . . . . . . . . . . . . 318 14.4 The Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 14.5 The Gelfand Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 15 C*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 15.1 Normal Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 15.2 Normal Operators in BðHÞ. . . . . . . . . . . . . . . . . . . . . . . . . . 353 15.3 The Spectral Theorem for Compact Normal Operators . . . . . . 360 15.4 Representation Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Hints to Selected Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Glossary of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415