# Book:N.L. Carothers/Real Analysis

## N.L. Carothers: *Real Analysis*

Published $2000$, **Cambridge University Press**

- ISBN 978-0-521-49756-5.

### Subject Matter

### Contents

- Preface

- PART ONE. METRIC SPACES

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

- 1 Calculus Review

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

- 2 Countable and Uncountable Sets

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

- 3 Metrics and Norms

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

- 4 Open Sets and Closed Sets

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

- 5 Continuity

- 6 Connectedness
- Connected Sets
- Notes and Remarks

- 6 Connectedness

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

- 7 Completeness

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

- 8 Compactness

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

- 9 Category

- PART TWO. FUNCTION SPACES

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

- 10 Sequences of Functions

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

- 11 The Space of Continuous Functions

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

- 12 The Stone–Weierstrass Theorem

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

- 13 Functions of Bounded Variation

- 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

- 14 The Riemann–Stieltjes Integral

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

- 15 Fourier Series

- 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

- 16 Lebesgue Measure

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

- 17 Measurable Functions

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

- 18 The Lebesgue Integral

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

- 19 Additional Topics

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

- 20 Differentiation

- References
- Symbol Index
- Topic Index