Real Analysis, in general deals with the convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity and interchange of limit operations. It teaches an understanding about theorems and helps in construction of proofs for the theorems. The principal goal of this book “Real Analysis, Part-1” is to provide the reader with a comprehensive knowledge of Real Analysis. This book is written in a user-friendly manner to make the material easy to understand by the reader. The proofs of the theorems are written clearly and informally as possible. The book has been designed to cater the need and requirement of Undergraduate, Postgraduate and Doctoral students of science, and engineering discipline in University and Colleges. This Real Analysis, Part 1-2 book is intended to provide students, professors, school teachers, technicians, researchers, scientists, mathematicians and engineers with a readily available reference to the essential theorems and their proofs. This book will contribute significantly to academic teaching, scientific research& practical works and it will enormously benefit the students for their preparation of all competitive exams, NET Exams, SLET/SET Exams, UPSC Exams, State Public Service Commission Exams and University Entrance Exam for admission to M.Sc. and Ph.D. programmes in Mathematics. Table of Contents Real Analysis (Part - I) Unit – 1 Introduction of Numbers 1.1 Introduction 1.2 Basic concepts in logic 1.3 Set Theory 1.4 Techniques for proofs 1.5 Number System 1.6 Metric spaces Unit – 2 Sequences and Series 2.1 Introduction 2.2 Sequences 2.3 Convergence of sequences 2.4 Subsequences and Cauchy sequence 2.5 Series 2.6 Power series Unit – 3 Limits and Continuity 3.1 Introduction 3.2 Limit of functions 3.3 Continuity of functions 3.4 Compactness and continuity 3.5 Connectedness and Continuity 3.6 Discontinuities Unit – 4 Sequences and Series of functions 4.1 Introduction 4.2 Pointwise convergence and Uniform convergence 4.3 Uniform convergence and continuity 4.4 Uniform convergence and integration 4.5 Uniform convergence and differentiation 4.6 Equicontinuous families of functions 4.7 Algebra of functions 4.8 The Stone-Weierstrass theorem 4.9 Problems Unit – 5 Differentiation of functions 5.1 Introduction 5.2 Derivatives of functions 5.3 Derivatives of higher order 5.4 Differentiation of vector valued functions Unit – 6 Integration - RS integrals 6.1 Introduction 6.2 Definition and existence of the integral 6.3 Properties of Riemann – Steiltjes integral 6.4 Integration and differentiation 6.5 Integration of vector valued functions 6.6 Rectifiable curves 6.7 Problems Unit – 7 Different Special Functions 7.1 Introduction 7.2 Power series 7.3 Exponential and logarithmic functions 7.4 Trigonometric functions 7.5 The algebraic completeness of the complex field 7.6 Fourier Series 7.7 Gamma function Unit – 8 Functions of Several Variables – Differentiation 8.1 Introduction 8.2 Linear transformations 8.3 Differentiation 8.4 Partial Derivatives 8.5 Contraction principle 8.6 Inverse function theorem 8.7 Implicit function theorem 8.8 The Projection 8.9 Rank theorem 8.10 Derivatives of higher order 8.11 Problems Unit – 9 Functions of Several Variables – Integration 9.1 Introduction 9.2 Iterated Integrals 9.3 Primitive Mapping 9.4 Change of variables 9.5 Partition of unity 9.6 Differentiation of integrals References Real Analysis (Part - 2) Unit – 1 Families of Sets 1–20 1.1 Introduction 1 1.2 Notations and Preliminaries 1 1.3 Algebras and σ-algebras 2 1.4 Measures 3 1.5 Outer Measures 8 1.6 Lebesgue outer measure 13 Unit – 2 Measurable Functions and Convergence Theorems 2.1 Introduction 2.2 Measurability 2.3 Types of Convergence 2.4 Lebesgue integral of Non - negative simple functions 2.5 Lebesgue integral of non - negative measurable functions 2.6 Lebesgue Integral of complex functions Unit – 3 Signed Measures and Radon Nikodym Theorem 3.1 Introduction 3.2 Positive and Negative sets 3.3 Hahn Decomposition theorem 3.4 Jordon Decomposition theorem 3.5 The Radon - Nikodym theorem 3.6 Complex Measures 3.7 Functions of Bounded Variations Unit – 4 Concepts of Functional Analysis 4.1 Introduction 4.2 Bounded Linear Functionals 4.3 Elementary Topology 4.4 Hilbert Space Unit – 5 LP Spaces and some Inequalities 5.1 Introduction 5.2 Fundamental concepts of Lp spaces 5.3 Some important inequalities 5.4 Convergence and completeness Unit – 6 Integration - RS Integrals 6.1 K – surfaces and K - forms 6.2 Simplexes and Chain 6.3 Stoke’s theorem 6.4 Vector field Unit – 7 Radon Measures 7.1 Introduction 7.2 Bounded linear functionals on Lp spaces 7.3 Riesz Representation theorem Unit – 8 Elements of Fourier Analysis 8.1 Introduction 8.2 The Convolutions 8.3 The Fourier Series 8.4 Gamma Function References