Real Analysis:

This text book is designed for an undergraduate course on mathematics. It covers the basic material that every graduate student should know in the classical theory of functions of real variables, measures, limits and continuity. This text book offers readability, practicality and flexibility. It presents fundamental theorems and ideas from a practical viewpoint, showing students the motivation behind mathematics and enabling them to construct their own proofs. Cover Contents Preface About the Author Chapter 1: Basic Properties of the Real Number System 1.1 Introduction 1.2 Order Structure of the Real Number System 1.3 Real Numbers and Decimal Expansions 1.4 The Extended Real Number System 1.5 Complex Field 1.6 The Euclidean Spaces Solved Exercises Unsolved Exercises Chapter 2: Some Finer Aspects of Set Theory 2.1 Introduction 2.2 Russel’s Paradox 2.3 Axiom of Choice 2.4 Sequences, Finite and Infinite Sets 2.5 Countable and Uncountable Sets 2.6 Cantor’s Inequality 2.7 Continuum Hypothesis Solved Exercises Unsolved Exercises Chapter 3: Sequences and Series 3.1 Introduction 3.2 Concepts Connected with Sequences 3.3 Basic Properties of Sequences and Series 3.4 Algebra of Series 3.5 Rearrangement of Series Solved Exercises Unsolved Exercises Chapter 4: Topological Aspects of the Real Line 4.1 Introduction 4.2 The Notion of Distance and the Idea of a Metric Space 4.3 Generalizations Solved Exercises Unsolved Exercises Chapter 5: Limits and Continuity 5.1 Introduction 5.2 Limits 5.3 Continuity 5.4 Discontinuities 5.5 Monotonic Functions 5.6 Uniform Continuity 5.7 Exponents 5.8 Generalizations Solved Exercises Unsolved Exercises Chapter 6: Differentiation 6.1 Introduction 6.2 Definition of Derivative, Examples and Arithmetic Rules 6.2.1 Arithmetic Rules 6.3 Local Extrema and Meanvalue Theorems 6.4 Taylor’s Theorem 6.5 L’Hospital’s Rule Solved Exercises Unsolved Exercises Chapter 7: Functions of Bounded Variation 7.1 Introduction 7.2 Definition and Examples 7.3 Properties of Total Variation 7.4 Functions of Bounded Variation and Monotonic Functions 7.5 Rectifiable Curves 7.6 Absolute Continuity 7.7 Generalizations Solved Exercises Unsolved Exercises Chapter 8: Riemann Integration 8.1 Introduction 8.2 Definition of the Riemann Integral and Examples 8.3 Properties of Riemann Integrals 8.4 Riemann Sums 8.5 Properties of Riemann Integrals 8.6 Meanvalue Theorems for Integral Calculus and the Rule for Change of Variable 8.7 Improper Integrals 8.8 Generalizations Solved Exercises Unsolved Exercises Chapter 9: Sequences and Series of Functions 9.1 Introduction 9.2 Pointwise Convergence, Bounded Convergence and Uniform Convergence 9.3 Properties 9.4 Families of Functions 9.5 Generalizations Solved Exercises Unsolved Exercises Chapter 10: Power Series and Special Functions 10.1 Introduction 10.2 Power Series 10.3 Exponential, Logarithm and Trigonometric Functions 10.4 Beta and Gamma Functions 10.5 Generalizations Solved Exercises Unsolved Exercises Chapter 11: Fourier Series 11.1 Introduction 11.2 Definitions and Examples Solved Exercises Unsolved Exercises Chapter 12: Real-Valued Functions of Two Real Variables 12.1 Introduction 12.2 Limits and Continuity 12.3 Differentiability 12.4 Higher Order Partial Derivatives 12.5 Extreme Values for a Function of Two Variables 12.6 Integration of Functions of Two Real Variables 12.7 Double Integrals 12.8 Generalizations Solved Exercises Unsolved Exercises Chapter 13: Lebesgue Measure Andintegration 13.1 Introduction 13.2 Outer Measure and Measurable Sets 13.2.1 Measurable Sets 13.3 Measurable Functions 13.4 Lebesgue Integral 13.5 Integration of Real-Valued Functions 13.6 Generalizations Solved Exercises Unsolved Exercises Chapter 14: Lp-Spaces 14.1 Introduction 14.2 Definitions and Examples 14.3 Properties of Lp -Spaces 14.4 Fourier Series on L1 [−π, π] and L2 [−π, π] 14.5 Generalizations Solved Exercises Unsolved Exercises Bibliography Index Real Analysis is designed for an undergraduate course on mathematics. It covers the basic material that every graduate student should know in the classical theory of functions of real variables, measures, limits and continuity. This text book offers readability, practicality and flexibility. It presents fundamental theorems and ideas from a practical viewpoint, showing students the motivation behind mathematics and enabling them to construct their own proofs