This book provides an introduction to basic topics in Real Analysis and makes the subject easily understandable to all learners. The book is useful for those that are involved with Real Analysis in disciplines such as mathematics, engineering, technology, and other physical sciences. It provides a good balance while dealing with the basic and essential topics that enable the reader to learn the more advanced topics easily. It includes many examples and end of chapter exercises including hints for solutions in several critical cases. The book is ideal for students, instructors, as well as those doing research in areas requiring a basic knowledge of Real Analysis. Those more advanced in the field will also find the book useful to refresh their knowledge of the topic. FeaturesIncludes basic and essential topics of real analysisAdopts a reasonable approach to make the subject easier to learnContains many solved examples and exercise at the end of each chapterPresents a quick review of the fundamentals of set theoryCovers the real number systemDiscusses the basic concepts of metric spaces and complete metric spaces
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1 | (12) |
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1.1 Introduction and Notations |
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1 | (1) |
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1.2 Ordered Pairs and Cartesian Product |
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2 | (1) |
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1.3 Relations and Functions |
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2 | (2) |
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1.4 Countable and Uncountable Sets |
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4 | (2) |
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6 | (3) |
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9 | (4) |
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13 | (12) |
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13 | (1) |
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14 | (1) |
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2.3 Geometrical Representation of Real Numbers and Intervals |
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15 | (1) |
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2.4 Integers, Rational Numbers, and Irrational Numbers |
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16 | (1) |
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2.5 Upper Bounds, Least Upper Bound or Supremum, the Completeness Axiom, Archimedean Property of R |
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16 | (2) |
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2.6 Infinite Decimal Representation of Real Numbers |
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18 | (2) |
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2.7 Absolute Value, Triangle Inequality, Cauchy--Schwarz Inequality |
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20 | (3) |
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2.8 Extended Real Number System R* |
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23 | (1) |
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24 | (1) |
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3 Sequences and Series of Real Numbers |
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25 | (20) |
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3.1 Convergent and Divergent Sequences of Real Numbers |
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25 | (1) |
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3.2 Limit Superior and Limit Inferior of a Sequence of Real Numbers |
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26 | (2) |
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3.3 Infinite Series of Real Numbers |
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28 | (6) |
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3.4 Convergence Tests for Infinite Series |
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34 | (3) |
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3.5 Rearrangements of Series |
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37 | (1) |
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3.6 Riemann's Theorem on Conditionally Convergent Series of Real Numbers |
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38 | (1) |
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3.7 Cauchy Multiplications of Series |
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39 | (2) |
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41 | (4) |
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4 Metric Spaces -- Basic Concepts, Complete Metric Spaces |
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45 | (16) |
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4.1 Metric and Metric Spaces |
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45 | (1) |
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4.2 Point Set Topology in Metric Spaces |
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46 | (7) |
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4.3 Convergent and Divergent Sequences in a Metric Space |
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53 | (1) |
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4.4 Cauchy Sequences and Complete Metric Spaces |
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54 | (2) |
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56 | (5) |
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61 | (38) |
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5.1 The Limit of Functions |
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61 | (2) |
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63 | (3) |
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5.3 Right-Hand and Left-Hand Limits |
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66 | (3) |
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5.4 Infinite Limits and Limits at Infinity |
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69 | (1) |
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5.5 Certain Important Limits |
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70 | (1) |
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5.6 Sequential Definition of Limit of a Function |
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71 | (1) |
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5.7 Cauchy's Criterion for Finite Limits |
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72 | (1) |
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73 | (2) |
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5.9 The Four Functional Limits at a Point |
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75 | (1) |
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5.10 Continuous and Discontinuous Functions |
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75 | (5) |
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5.11 Some Theorems on the Continuity |
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80 | (3) |
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5.12 Properties of Continuous Functions |
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83 | (2) |
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85 | (3) |
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5.14 Continuity and Uniform Continuity in Metric Spaces |
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88 | (3) |
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91 | (8) |
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6 Connectedness and Compactness |
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99 | (24) |
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99 | (6) |
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6.2 The Intermediate Value Theorem |
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105 | (2) |
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107 | (1) |
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108 | (6) |
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6.5 The Finite Intersection Property |
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114 | (2) |
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6.6 The Heine--Borel Theorem |
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116 | (4) |
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120 | (3) |
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123 | (34) |
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123 | (3) |
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7.2 The Differential Calculus |
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126 | (6) |
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7.3 Properties of Differentiable Functions |
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132 | (6) |
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138 | (9) |
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147 | (7) |
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154 | (3) |
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157 | (56) |
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157 | (11) |
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8.2 Properties of the Riemann Integral |
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168 | (6) |
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8.3 The Fundamental Theorems of Calculus |
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174 | (5) |
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8.4 The Substitution Theorem and Integration by Parts |
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179 | (2) |
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181 | (6) |
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8.6 The Riemann--Stieltjes Integral |
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187 | (9) |
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8.7 Functions of Bounded Variation |
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196 | (9) |
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205 | (8) |
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9 Sequences and Series of Functions |
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213 | (22) |
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9.1 The Pointwise Convergence of Sequences of Functions and the Uniform Convergence |
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213 | (2) |
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9.2 The Uniform Convergence and the Continuity, the Cauchy Criterion for the Uniform Convergence |
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215 | (2) |
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9.3 The Uniform Convergence of Infinite Series of Functions |
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217 | (2) |
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9.4 The Uniform Convergence of Integrations and Differentiations |
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219 | (3) |
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9.5 The Equicontinuous Family of Functions and the Arzela-Ascoli Theorem |
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222 | (2) |
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9.6 Dirichlet's Test for the Uniform Convergence |
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224 | (1) |
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9.7 The Weierstrass Theorem |
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225 | (2) |
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227 | (5) |
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232 | (3) |
| Bibliography |
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235 | (2) |
| Index |
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237 | |
Hemen Dutta is a faculty member at the Department of Mathematics, Gauhati University, India. He did his Master of Science in Mathematics, Post Graduate Diploma in Computer Application and Ph.D. in Mathematics from Gauhati University, India. He received his M.Phil in Mathematics from Madurai Kamaraj University, India. His primary research interest includes areas of mathematical analysis. He has to his credit several research papers, some book chapters and few books. He has delivered talks at different institutions and organized a number of academic events. He is a member of several mathematical societies. He has also been a resource person in training programmes for school students and teachers and delivered popular talks. He has authored several newspaper and popular articles in science education, research, etc. He has served the Assam Academy of Mathematics as joint-secretary (honorary) for two years.
P.N. Natarajan is Dr. Radhakrishnan Awardee for the Best Teacher in Mathematics for the year 1990-91 by the Govt. of TamilNadu, India. He has been working as an independent researcher in Mathematics since his retirement in 2004, as Professor and Head, Dept. of Mathematics, Ramakrishna Mission Vivekananda College, Chennai, India. He received his Ph.D. in Analysis from the University of Madras in 1980. He has to his credit over 100 research papers published in several reputed national and international journals. He has authored 3 books (one of them 2 editions) and contributed a chapter to each of 2 edited volumes, published by Springer, Taylor-Francis and Wiley. Dr. Natarajan's research interest includes Summability Theory and Functional Analysis, both Classical and Ultrametric. Besides visiting institutes of repute in Canada, France, Holland and Greece on invitation, he has participated in several international conferences and has chaired sessions.
Yeol Je Cho is an Emeritus Professor at Department of Mathematics Education, Gyeongsang National University, Jinju, Korea, and a Distinguished Professor at School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, China. Also, he is a Fellow of the Korean Academy of Science and Technology, Seoul, Korea, since 2006, and a member of several mathematical Societies. He received his Bachelor´s Degree, Masters Degree, Ph.D. in Mathematics from Pusan National University, Pusan, Korea, and, in 1988, he was a Post-Doctor Fellow in Department of Mathematics, Saint Louis University, Saint Louis, USA. He is an Organizer of the International Conference on Nonlinear Functional Analysis and Applications, some Workshops and Symposiums on Nonlinear Analysis and Applications and Members of the Editorial Boards of 10 more International Journals of Mathematics. He has published 350 more papers and some book chapters, 20 more Monographs and 10 more Books in CRC, Taylor & Francis, Springer, Nova Science Publishers, New York, USA. His research areas are Nonlinear Analysis and Applications, especially, fixed point theory and applications, some kinds of nonlinear problems, that is, equilibrium problems, variational inequality problems, saddle point problems, optimization problems, inequality theory and applications, stability of functional equations and applications. He delivered many Invited Talks at many International Conferences on Nonlinear Analysis and Applications, which have been opened in many countries.
Lisainfo e-raamatute kohta