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.
Features
Includes basic and essential topics of real analysis Adopts a reasonable approach to make the subject easier to learn Contains many solved examples and exercise at the end of each chapter Presents a quick review of the fundamentals of set theory Covers the real number system Discusses the basic concepts of metric spaces and complete metric spaces
Author(s): Hemen Dutta, Pinnangudi N. Natarajan, Yeol Je Cho
Publisher: CRC Press
Year: 2020
Language: English
Pages: xii+240
Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Authors
1 Review of Set Theory
1.1 Introduction and Notations
1.2 Ordered Pairs and Cartesian Product
1.3 Relations and Functions
1.4 Countable and Uncountable Sets
1.5 Set Algebras
1.6 Exercises
2 The Real Number System
2.1 Field Axioms
2.2 Order Axioms
2.3 Geometrical Representation of Real Numbers and Intervals
2.4 Integers, Rational Numbers, and Irrational Numbers
2.5 Upper Bounds, Least Upper Bound or Supremum, the Completeness Axiom, Archimedean Property of
2.6 Infinite Decimal Representation of Real Numbers
2.7 Absolute Value, Triangle Inequality, Cauchy-Schwarz Inequality
2.8 Extended Real Number System R*
2.9 Exercises
3 Sequences and Series of Real Numbers
3.1 Convergent and Divergent Sequences of Real Numbers
3.2 Limit Superior and Limit Inferior of a Sequence of Real Numbers
3.3 Infinite Series of Real Numbers
3.4 Convergence Tests for Infinite Series
3.5 Rearrangements of Series
3.6 Riemann's Theorem on Conditionally Convergent Series of Real Numbers
3.7 Cauchy Multiplications of Series
3.8 Exercises
4 Metric Spaces - Basic Concepts, Complete Metric Spaces
4.1 Metric and Metric Spaces
4.2 Point Set Topology in Metric Spaces
4.3 Convergent and Divergent Sequences in a Metric Space
4.4 Cauchy Sequences and Complete Metric Spaces
4.5 Exercises
5 Limits and Continuity
5.1 The Limit of Functions
5.2 Algebras of Limits
5.3 Right-Hand and Left-Hand Limits
5.4 Infinite Limits and Limits at Infinity
5.5 Certain Important Limits
5.6 Sequential Definition of Limit of a Function
5.7 Cauchy's Criterion for Finite Limits
5.8 Monotonic Functions
5.9 The Four Functional Limits at a Point
5.10 Continuous and Discontinuous Functions
5.11 Some Theorems on the Continuity
5.12 Properties of Continuous Functions
5.13 Uniform Continuity
5.14 Continuity and Uniform Continuity in Metric Spaces
5.15 Exercises
6 Connectedness and Compactness
6.1 Connectedness
6.2 The Intermediate Value Theorem
6.3 Components
6.4 Compactness
6.5 The Finite Intersection Property
6.6 The Heine-Borel Theorem
6.7 Exercises
7 Differentiation
7.1 The Derivative
7.2 The Differential Calculus
7.3 Properties of Differentiable Functions
7.4 The L'Hospital Rule
7.5 Taylor's Theorem
7.6 Exercises
8 Integration
8.1 The Riemann Integral
8.2 Properties of the Riemann Integral
8.3 The Fundamental Theorems of Calculus
8.4 The Substitution Theorem and Integration by Parts
8.5 Improper Integrals
8.6 The Riemann-Stieltjes Integral
8.7 Functions of Bounded Variation
8.8 Exercises
9 Sequences and Series of Functions
9.1 The Pointwise Convergence of Sequences of Functions and the Uniform Convergence
9.2 The Uniform Convergence and the Continuity, the Cauchy Criterion for the Uniform Convergence
9.3 The Uniform Convergence of Infinite Series of Functions
9.4 The Uniform Convergence of Integrations and Differentiations
9.5 The Equicontinuous Family of Functions and the Arzela-Ascoli Theorem
9.6 Dirichlet's Test for the Uniform Convergence
9.7 The Weierstrass Theorem
9.8 Some Examples
9.9 Exercises
Bibliography
Index
