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Texts and Readings in Mathematics 37 Terence Tao Analysis I Third Edition Texts and Readings in Mathematics Volume 37 Advisory Editor C.S. Seshadri, Chennai Mathematical Institute, Chennai Managing Editor Rajendra Bhatia, Indian Statistical Institute, New Delhi Editor Manindra Agrawal, Indian Institute of Technology Kanpur, Kanpur V. Balaji, Chennai Mathematical Institute, Chennai R.B. Bapat, Indian Statistical Institute, New Delhi V.S. Borkar, Indian Institute of Technology Bombay, Mumbai T.R. Ramadas, Chennai Mathematical Institute, Chennai V. Srinivas, Tata Institute of Fundamental Research, Mumbai The Texts and Readings in Mathematics series publishes high-quality textbooks, research-level monographs, lecture notes and contributed volumes. Undergraduate and graduate students of mathematics, research scholars, and teachers would find this book series useful. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. The books in this series are co-published with Hindustan Book Agency, New Delhi, India. More information about this series at http://www.springer.com/series/15141 Terence Tao Analysis I Third Edition 123 Terence Tao Department ofMathematics University of California, LosAngeles LosAngeles, CA USA Thisworkisaco-publicationwithHindustanBookAgency,NewDelhi,licensedforsaleinall countriesinelectronicformonly.SoldanddistributedinprintacrosstheworldbyHindustan BookAgency,P-19GreenParkExtension,NewDelhi110016,India.ISBN:978-93-80250-64-9 ©HindustanBookAgency2015. ISSN 2366-8725 (electronic) TextsandReadings inMathematics ISBN978-981-10-1789-6 (eBook) DOI 10.1007/978-981-10-1789-6 LibraryofCongressControlNumber:2016940817 ©SpringerScience+BusinessMediaSingapore2016andHindustanBookAgency2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublishers,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. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublishers,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis book are believed to be true and accurate at the date of publication. Neither the publishers nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerScience+BusinessMediaSingaporePteLtd. To my parents, for everything Contents Preface to the second and third editions xi Preface to the first edition xiii About the Author xix 1 Introduction 1 1.1 What is analysis? . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Why do analysis? . . . . . . . . . . . . . . . . . . . . . . 2 2 Starting at the beginning: the natural numbers 13 2.1 The Peano axioms . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 29 3 Set theory 33 3.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Russell’s paradox (Optional) . . . . . . . . . . . . . . . . 46 3.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 Images and inverse images . . . . . . . . . . . . . . . . . 56 3.5 Cartesian products . . . . . . . . . . . . . . . . . . . . . 62 3.6 Cardinality of sets . . . . . . . . . . . . . . . . . . . . . . 67 4 Integers and rationals 74 4.1 The integers . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 The rationals . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 Absolute value and exponentiation . . . . . . . . . . . . . 86 4.4 Gaps in the rational numbers . . . . . . . . . . . . . . . . 90 5 The real numbers 94 5.1 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . 96 5.2 Equivalent Cauchy sequences . . . . . . . . . . . . . . . . 100 5.3 The construction of the real numbers . . . . . . . . . . . 102 5.4 Ordering the reals . . . . . . . . . . . . . . . . . . . . . . 111 vii viii Contents 5.5 The least upper bound property . . . . . . . . . . . . . . 116 5.6 Real exponentiation, part I . . . . . . . . . . . . . . . . . 121 6 Limits of sequences 126 6.1 Convergence and limit laws . . . . . . . . . . . . . . . . . 126 6.2 The Extended real number system . . . . . . . . . . . . . 133 6.3 Suprema and Infima of sequences . . . . . . . . . . . . . 137 6.4 Limsup, Liminf, and limit points . . . . . . . . . . . . . . 139 6.5 Some standard limits . . . . . . . . . . . . . . . . . . . . 148 6.6 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.7 Real exponentiation, part II . . . . . . . . . . . . . . . . 152 7 Series 155 7.1 Finite series . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.2 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.3 Sums of non-negative numbers . . . . . . . . . . . . . . . 170 7.4 Rearrangement of series . . . . . . . . . . . . . . . . . . . 174 7.5 The root and ratio tests . . . . . . . . . . . . . . . . . . . 178 8 Infinite sets 181 8.1 Countability . . . . . . . . . . . . . . . . . . . . . . . . . 181 8.2 Summation on infinite sets . . . . . . . . . . . . . . . . . 188 8.3 Uncountable sets. . . . . . . . . . . . . . . . . . . . . . . 195 8.4 The axiom of choice . . . . . . . . . . . . . . . . . . . . . 198 8.5 Ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . 202 9 Continuous functions on R 211 9.1 Subsets of the real line . . . . . . . . . . . . . . . . . . . 211 9.2 The algebra of real-valued functions . . . . . . . . . . . . 217 9.3 Limiting values of functions . . . . . . . . . . . . . . . . 220 9.4 Continuous functions . . . . . . . . . . . . . . . . . . . . 227 9.5 Left and right limits . . . . . . . . . . . . . . . . . . . . . 231 9.6 The maximum principle . . . . . . . . . . . . . . . . . . . 234 9.7 The intermediate value theorem . . . . . . . . . . . . . . 238 9.8 Monotonic functions . . . . . . . . . . . . . . . . . . . . . 241 9.9 Uniform continuity . . . . . . . . . . . . . . . . . . . . . 243 9.10 Limits at infinity. . . . . . . . . . . . . . . . . . . . . . . 249 10 Differentiation of functions 251 10.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . 251 Contents ix 10.2 Local maxima, local minima, and derivatives . . . . . . . 257 10.3 Monotone functions and derivatives . . . . . . . . . . . . 260 10.4 Inverse functions and derivatives . . . . . . . . . . . . . . 261 10.5 L’Hˆopital’s rule . . . . . . . . . . . . . . . . . . . . . . . 264 11 The Riemann integral 267 11.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 11.2 Piecewise constant functions . . . . . . . . . . . . . . . . 272 11.3 Upper and lower Riemann integrals . . . . . . . . . . . . 276 11.4 Basic properties of the Riemann integral . . . . . . . . . 280 11.5 Riemann integrability of continuous functions . . . . . . 285 11.6 Riemann integrability of monotone functions . . . . . . . 289 11.7 A non-Riemann integrable function . . . . . . . . . . . . 291 11.8 The Riemann-Stieltjes integral . . . . . . . . . . . . . . . 292 11.9 The two fundamental theorems of calculus . . . . . . . . 295 11.10 Consequences of the fundamental theorems . . . . . . . . 300 A Appendix: the basics of mathematical logic 305 A.1 Mathematical statements . . . . . . . . . . . . . . . . . . 306 A.2 Implication . . . . . . . . . . . . . . . . . . . . . . . . . . 312 A.3 The structure of proofs . . . . . . . . . . . . . . . . . . . 317 A.4 Variables and quantifiers . . . . . . . . . . . . . . . . . . 320 A.5 Nested quantifiers . . . . . . . . . . . . . . . . . . . . . . 324 A.6 Some examples of proofs and quantifiers . . . . . . . . . 327 A.7 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 B Appendix: the decimal system 331 B.1 The decimal representation of natural numbers . . . . . . 332 B.2 The decimal representation of real numbers . . . . . . . . 335 Index 339 Texts andReadingsinMathematics 349 Preface to the second and third editions Since the publication of the first edition, many students and lectur- ers have communicated a number of minor typos and other corrections to me. There was also some demand for a hardcover edition of the texts. Because of this, the publishers and I have decided to incorporate the corrections and issue a hardcover second edition of the textbooks. The layout, page numbering, and indexing of the texts have also been changed; in particular the two volumes are now numbered and indexed separately. However, the chapter and exercise numbering, as well as the mathematical content, remains the same as the first edition, and so the two editions can be used more or less interchangeably for homework and study purposes. Thethirdeditioncontainsanumberofcorrectionsthatwerereported forthesecondedition,togetherwithafewnewexercises,butisotherwise essentially the same text. xi

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