Real Analysis

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