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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 of Mathematics University of California, Los Angeles Los Angeles, CA USA This work is a co-publication with Hindustan Book Agency, New Delhi, licensed for sale in all countries in electronic form only Sold and distributed in print across the world by Hindustan Book Agency, P-19 Green Park Extension, New Delhi 110016, India ISBN: 978-93-80250-64-9 © Hindustan Book Agency 2015 ISSN 2366-8725 (electronic) Texts and Readings in Mathematics ISBN 978-981-10-1789-6 (eBook) DOI 10.1007/978-981-10-1789-6 Library of Congress Control Number: 2016940817 © Springer Science+Business Media Singapore 2016 and Hindustan Book Agency 2015 This work is subject to copyright All rights are reserved by the Publishers, whether the whole or part 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 or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publishers, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made This Springer imprint is published by Springer Nature The registered company is Springer Science+Business Media Singapore Pte Ltd To my parents, for everything Contents Preface to the second and third editions xi Preface to the first edition xiii About the Author xix Introduction 1.1 What is analysis? 1.2 Why analysis? Starting at the beginning: 2.1 The Peano axioms 2.2 Addition 2.3 Multiplication the natural Set theory 3.1 Fundamentals 3.2 Russell’s paradox (Optional) 3.3 Functions 3.4 Images and inverse images 3.5 Cartesian products 3.6 Cardinality of sets 1 numbers 13 15 24 29 33 33 46 49 56 62 67 Integers and rationals 4.1 The integers 4.2 The rationals 4.3 Absolute value and exponentiation 4.4 Gaps in the rational numbers 74 74 81 86 90 real numbers Cauchy sequences Equivalent Cauchy sequences The construction of the real numbers Ordering the reals 94 96 100 102 111 The 5.1 5.2 5.3 5.4 vii viii Contents 5.5 5.6 The least upper bound property 116 Real exponentiation, part I 121 Limits of sequences 6.1 Convergence and limit laws 6.2 The Extended real number system 6.3 Suprema and Infima of sequences 6.4 Limsup, Liminf, and limit points 6.5 Some standard limits 6.6 Subsequences 6.7 Real exponentiation, part II 126 126 133 137 139 148 149 152 Series 7.1 Finite series 7.2 Infinite series 7.3 Sums of non-negative numbers 7.4 Rearrangement of series 7.5 The root and ratio tests 155 155 164 170 174 178 Infinite sets 8.1 Countability 8.2 Summation on infinite sets 8.3 Uncountable sets 8.4 The axiom of choice 8.5 Ordered sets 181 181 188 195 198 202 Continuous functions on R 9.1 Subsets of the real line 9.2 The algebra of real-valued functions 9.3 Limiting values of functions 9.4 Continuous functions 9.5 Left and right limits 9.6 The maximum principle 9.7 The intermediate value theorem 9.8 Monotonic functions 9.9 Uniform continuity 9.10 Limits at infinity 211 211 217 220 227 231 234 238 241 243 249 10 Differentiation of functions 251 10.1 Basic definitions 251 ix Contents 10.2 10.3 10.4 10.5 Local maxima, local minima, and derivatives Monotone functions and derivatives Inverse functions and derivatives L’Hˆopital’s rule 11 The Riemann integral 11.1 Partitions 11.2 Piecewise constant functions 11.3 Upper and lower Riemann integrals 11.4 Basic properties of the Riemann integral 11.5 Riemann integrability of continuous functions 11.6 Riemann integrability of monotone functions 11.7 A non-Riemann integrable function 11.8 The Riemann-Stieltjes integral 11.9 The two fundamental theorems of calculus 11.10 Consequences of the fundamental theorems A Appendix: the basics of mathematical logic A.1 Mathematical statements A.2 Implication A.3 The structure of proofs A.4 Variables and quantifiers A.5 Nested quantifiers A.6 Some examples of proofs and quantifiers A.7 Equality 257 260 261 264 267 268 272 276 280 285 289 291 292 295 300 305 306 312 317 320 324 327 329 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 and Readings in Mathematics 349 Preface to the second and third editions Since the publication of the first edition, many students and lecturers 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 The third edition contains a number of corrections that were reported for the second edition, together with a few new exercises, but is otherwise essentially the same text xi ... 11 The Riemann integral 11 .1 Partitions 11 .2 Piecewise constant functions 11 .3 Upper and lower Riemann integrals 11 .4 Basic properties... 211 211 217 220 227 2 31 234 238 2 41 243 249 10 Differentiation of functions 2 51 10 .1 Basic definitions 2 51 ix Contents 10 .2 10 .3 10 .4 10 .5 Local maxima, local minima,... 94 96 10 0 10 2 11 1 The 5 .1 5.2 5.3 5.4 vii viii Contents 5.5 5.6 The least upper bound property 11 6 Real exponentiation, part I 12 1 Limits of
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