Book:N.L. Carothers/Real Analysis

Subject Matter

 * Real Analysis

Contents

 * Preface


 * PART ONE. METRIC SPACES


 * 1 Calculus Review
 * The Real Numbers
 * Limits and Continuity
 * Notes and Remarks


 * 2 Countable and Uncountable Sets
 * Equivalence and Cardinality
 * The Cantor Set
 * Monotone Functions
 * Notes and Remarks


 * 3 Metrics and Norms
 * Metric Spaces
 * Normed Vector Spaces
 * More Inequalities
 * Limits in Metric Spaces
 * Notes and Remarks


 * 4 Open Sets and Closed Sets
 * Open Sets
 * Closed Sets
 * The Relative Metric
 * Notes and Remarks


 * 5 Continuity
 * Continuous Functions
 * Homeomorphisms
 * The Space of Continuous Functions


 * 6 Connectedness
 * Connected Sets
 * Notes and Remarks


 * 7 Completeness
 * Totally Bounded Sets
 * Complete Metric Spaces
 * Fixed Points
 * Completions
 * Notes and Remarks


 * 8 Compactness
 * Compact Metric Spaces
 * Uniform Continuity
 * Equivalent Metrics
 * Notes and Remarks


 * 9 Category
 * Discontinuous Functions
 * The Baire Category Theorem
 * Notes and Remarks


 * PART TWO. FUNCTION SPACES


 * 10 Sequences of Functions
 * Historical Background
 * Pointwise and Uniform Convergence
 * Interchanging Limits
 * The Space of Bounded Functions
 * Notes and Remarks


 * 11 The Space of Continuous Functions
 * The Weierstrass Theorem
 * Trigonometric Polynomials
 * Infinitely Differentiable Functions
 * Equicontinuity
 * Continuity and Category
 * Notes and Remarks


 * 12 The Stone–Weierstrass Theorem
 * Algebras and Lattices
 * The Stone–Weierstrass Theorem
 * Notes and Remarks


 * 13 Functions of Bounded Variation
 * Functions of Bounded Variation
 * Helly's First Theorem
 * Notes and Remarks


 * 14 The Riemann–Stieltjes Integral
 * Weights and Measures
 * The Riemann–Stieltjes Integral
 * The Space of Integrable Functions
 * Integrators of Bounded Variation
 * The Riemann Integral
 * The Riesz Representation Theorem
 * Other Definitions, Other Properties
 * Notes and Remarks


 * 15 Fourier Series
 * Preliminaries
 * Dirichlet's Formula
 * Fejér's Theorem
 * Complex Fourier Series
 * Notes and Remarks


 * PART THREE. LEBESGUE MEASURE AND INTEGRATION


 * 16 Lebesgue Measure
 * The Problem of Measure
 * Lebesgue Outer Measure
 * Riemann Integrability
 * Measurable Sets
 * The Structure of Measurable Sets
 * A Nonmeasurable Set
 * Other Definitions
 * Notes and Remarks


 * 17 Measurable Functions
 * Measurable Functions
 * Extended Real-Valued Functions
 * Sequences of Measurable Functions
 * Approximation of Measurable Functions
 * Notes and Remarks


 * 18 The Lebesgue Integral
 * Simple Functions
 * Nonnegative Functions
 * The General Case
 * Lebesgue's Dominated Convergence Theorem
 * Approximation of Integrable Functions
 * Notes and Remarks


 * 19 Additional Topics
 * Convergence in Measure
 * The $L_p$ Spaces
 * Approximation of $L_p$ Functions
 * More on Fourier Series
 * Notes and Remarks


 * 20 Differentiation
 * Lebesgue's Differentiation Theorem
 * Absolute Continuity
 * Notes and Remarks


 * References
 * Symbol Index
 * Topic Index