logo资料库

Analysis I Terence Tao third edition.pdf

发布时间：2022-06-13 发布人：admin 分类：说明书 资料大小：2.52M 资料格式：pdf 举报 版权申诉
第1页 / 共366页
第2页 / 共366页
第3页 / 共366页
第4页 / 共366页
第5页 / 共366页
第6页 / 共366页
第7页 / 共366页
第8页 / 共366页
资料共366页，剩余部分请下载后查看
    Contents
    Preface to the second and third editions
    Preface to the first edition
    About the Author
    1 Introduction
    1.1 What is analysis?
    1.2 Why do analysis?
    2 Starting at the beginning: the natural numbers
    2.1 The Peano axioms
    2.2 Addition
    2.3 Multiplication
    3 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
    4 Integers and rationals
    4.1 The integers
    4.2 The rationals
    4.3 Absolute value and exponentiation
    4.4 Gaps in the rational numbers
    5 The real numbers
    5.1 Cauchy sequences
    5.2 Equivalent Cauchy sequences
    5.3 The construction of the real numbers
    5.4 Ordering the reals
    5.5 The least upper bound property
    5.6 Real exponentiation, part I
    6 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
    7 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
    8 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
    9 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
    10 Differentiation of functions
    10.1 Basic definitions
    10.2 Local maxima, local minima, and derivatives
    10.3 Monotone functions and derivatives
    10.4 Inverse functions and derivatives
    10.5 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
    B Appendix: the decimal system
    B.1 The decimal representation of natural numbers
    B.2 The decimal representation of real numbers
    Index
    Texts and Readings in Mathematics
    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. (electronic) ISSN 2366-8725 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, 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. trademarks, service marks, etc. 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 Preface to the first edition About the Author 1 Introduction 1.1 What is analysis? . . . . . . . . . . . . . . . . . . . . . . 1.2 Why do analysis? . . . . . . . . . . . . . . . . . . . . . . xi xiii xix 1 1 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 Fundamentals 33 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Russell’s paradox (Optional) . . . . . . . . . . . . . . . . 46 3.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 . . . . . . . . . . . . . . . . . 56 Images and inverse images 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 . . . . . . . . . . . . . 137 Suprema and Infima of sequences Limsup, Liminf, and limit points . . . . . . . . . . . . . . 139 6.4 Some standard limits . . . . . . . . . . . . . . . . . . . . 148 6.5 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.6 6.7 Real exponentiation, part II . . . . . . . . . . . . . . . . 152 7 Series 155 Finite series . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.1 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.2 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 Limiting values of functions 211 9.1 Subsets of the real line . . . . . . . . . . . . . . . . . . . 211 9.2 The algebra of real-valued functions . . . . . . . . . . . . 217 . . . . . . . . . . . . . . . . 220 9.3 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
    分享到：
    赞 收藏

    相关推荐

    资料库

    课程资源

    共收录17145份资料，累计13个分类，关注成员有19位，主要包括：PHP,网络管理,网页制作,Java,.Net,数据库,3G/移动开发,C/C++,游戏开发,嵌入式,讲义,软件测试,专业指导

    热门标签

    PHP 网络管理 网页制作 Java .Net 数据库 3G/移动开发 C/C++ 游戏开发 嵌入式 讲义 软件测试 专业指导

    最新资料