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Texts and Readings in Mathematics 37 Terence Tao Analysis I Fourth Edition Texts and Readings in Mathematics Advisory Editor C S Seshadri Chennai Mathematical Institute Chennai India Managing Editor Rajendra Bhatia Ashoka University Sonepat India Editorial Board Manindra Agrawal Indian Institute of Technology Kanpur India V Balaji Chennai Mathematical Institute Chennai India R B Bapat Indian Statistical Institute New Delhi India V S Borkar Indian Institute of Technology Mumbai India Apoorva Khare Indian Institute of Science Bangalore India T R Ramadas Chennai Mathematical Institute Chennai India V Srinivas Tata Institute of Fundamental Research Mumbai India Technical Editor P Vanchinathan Vellore Institute of Technology Chennai India 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 nd this book series useful The volumes are carefully written as teaching aids and high light characteristic features of the theory Books in this series are co published with Hindustan Book Agency New Delhi India Terence Tao Analysis I Fourth Edition Terence Tao Department of Mathematics University of California Los Angeles Los Angeles CA USA ISSN 2366 8717 Texts and Readings in Mathematics ISBN https doi org 10 1007 978 981 19 7261 4 ISSN 2366 8725 electronic ISBN 978 981 19 7261 4 eBook Jointly published with Hindustan Book Agency 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 81 951961 9 7 Hindustan Book Agency 2022 Springer has only electronic right 3rd edition Springer Science Business Media Singapore 2016 and Hindustan Book Agency 2015 4th edition Hindustan Book Agency 2022 This work is subject to copyright All rights are solely and exclusively licensed by the Publisher whether the whole or part of the material is concerned speci cally the rights of translation reprinting reuse of illustrations recitation broadcasting reproduction on micro lms 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 speci c 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 expressed or implied with respect to the material contained herein or for any errors or omissions that may have been made The publishers remain neutral with regard to jurisdictional claims in published maps and institutional af liations This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd The registered company address is 152 Beach Road 21 01 04 Gateway East Singapore 189721 Singapore To my parents for everything Preface to the First Edition This text originated from the lecture notes I gave teaching the honours undergraduate level real analysis sequence at the University of California Los Angeles in 2003 Among the undergraduates here real analysis was viewed as being one of the most dif cult courses to learn not only because of the abstract concepts being introduced for the rst time e g topology limits measurability etc but also because of the level of rigour and proof demanded of the course Because of this perception of dif culty one was often faced with the dif cult choice of either reducing the level of rigour in the course in order to make it easier or to maintain strict standards and face the prospect of many undergraduates even many of the bright and enthusiastic ones struggling with the course material Faced with this dilemma I tried a somewhat unusual approach to the subject Typically an introductory sequence in real analysis assumes that the students are already familiar with the real numbers with mathematical induction with elementary calculus and with the basics of set theory and then quickly launches into the heart of the subject for instance the concept of a limit Normally students entering this sequence do indeed have a fair bit of exposure to these prerequisite topics though in most cases the material is not covered in a thorough manner For instance very few students were able to actually de ne a real number or even an integer properly even though they could visualize these numbers intuitively and manipulate them algebraically This seemed to me to be a missed opportunity Real analysis is one of the rst subjects together with linear algebra and abstract algebra that a student encounters in which one truly has to grapple with the subtleties of a truly rigorous mathematical proof As such the course offered an excellent chance to go back to the foundations of mathematics and in particular the opportunity to do a proper and thorough construction of the real numbers Thus the course was structured as follows In the rst week I described some well known paradoxes in analysis in which standard laws of the subject e g interchange of limits and sums or sums and integrals were applied in a non rigorous way to give nonsensical results such as 0 1 This motivated the need to go back to the very beginning of the subject even to the very de nition of the natural numbers and check all the foundations from scratch For instance one of the rst homework vii viii Preface to the First Edition assignments was to check using only the Peano axioms that addition was associative for natural numbers i e that a b c a b c for all natural numbers a b c see Exercise 2 2 1 Thus even in the rst week the students had to write rigorous proofs using mathematical induction After we had derived all the basic properties of the natural numbers we then moved on to the integers initially de ned as formal differences of natural numbers once the students had veri ed all the basic properties of the integers we moved on to the rationals initially de ned as formal

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