# Book:N.L. Carothers/Real Analysis

## N.L. Carothers: Real Analysis

Published $2000$, Cambridge University Press

ISBN 978-0-521-49756-5.

### 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
The $L_p$ Spaces
Approximation of $L_p$ Functions