Book:H.L. Royden/Real Analysis/Second Edition

H.L. Royden: Real Analysis (2nd Edition)

Published $\text {1968}$, Macmillan Publishing Company

ISBN 0-02-404150-5

Subject Matter

• Real Analysis

Contents

Prologue to the Student

1 Set Theory

1 Introduction

2 Functions

3 Unions, intersections, and complements

4 Algebras of sets

5 The axiom of choice and infinite direct products

6 Countable sets

7 Relations and equivalences

8 Partial orderings and the maximal principle

9 Well ordering and the countable ordinals

Part One Theory of Functions of a Real Variable

2 The Real Number System

1 Axioms for the real numbers

2 The natural and rational numbers as subsets of $\mathbf R$

3 The extended real numbers

4 Sequences of real numbers

5 Open and closed sets of real numbers

6 Continuous functions

7 Borel sets

3 Lebesgue Measure

1 Introduction

2 Outer measure

3 Measurable sets and Lebesgue measure

*4 A nonmeasurable set

5 Measurable functions

6 Littlewood's three principles

4 The Lebesgue Integral

1 The Riemann integral

2 The Lebesgue integral of a bounded function over a set of finite measure

3 The integral of a nonnegative function

4 The general Lebesgue integral

*5 Convergence in measure

5 Differentiation and Integration

1 Differentiation of monotone functions

2 Functions of bounded variation

3 Differentiation of an integral

4 Absolute continuity

*5 Convex functions

6 The Classical Banach Spaces

1 The $L^p$ spaces

2 The Minkowski and Hölder inequalities

3 Convergence and completeness

4 Approximation in $L^p$

5 Bounded linear functionals on the $L^p$ spaces

Part Two Abstract Spaces

7 Metric Spaces

1 Introduction

2 Open and closed sets

3 Continuous functions and homeomorphisms

4 Convergence and completeness

5 Uniform continuity and uniformity

6 Subspaces

8 Baire category

8 Topological Spaces

1 Fundamental notions

2 Bases and countability

3 The separation axioms and continuous real-valued functions

4 Product spaces

5 Connectedness

*6 Absolute $\mathcal G_\delta$'s

*7 Nets

9 Compact Spaces

1 Basic properties

2 Countable compactness and the Bolzano-Weierstrass property

3 Compact metric spaces

4 Products of compact spaces

4 Locally compact spaces

*6 The Stone-Čech compactification

7 The Stone-Weierstrass theorem

* The Ascoli theorem

10 Banach Spaces

1 Introduction

2 Linear operators

3 Linear functionals and the Hahn-Banach theorem

4 The closed graph theorem

*5 Topological vector spaces

*6 Weak topologies

*7 Convexity

8 Hilbert space

Part Three General Measure and Integration Theory

11 Measure and Integration

1 Measure spaces

2 Measurable functions

3 Integration

*4 General convergence theorems

5 Signed measures

6 The Radon-Nikodym Theorem

7 The $L^p$ spaces

12 Measure and Outer Measure

1 Outer measure and measurability

2 The extension theorem

*3 The Lebesgue-Stieltjes integral

4 Product measures

*5 Inner measure

*6 Extension by sets of measure zero

*7 Carathéodory outer measure

13 The Daniell Integral

1 Introduction

2 The extension theorem

3 Uniqueness

4 Measurability and measure

14 Measure and Topology

1 Baire sets and Borel sets

2 Positive linear functionals and Borel measures

3 Bounded linear functionals on $C \left({X}\right)$

*4 The Borel extension of a measure

15 Mappings of Measure Spaces

1 Point mappings and set mappings

2 Measure algebras

3 Borel equivalences

4 Set mappings and point mappings on complete separable metric spaces

5 The isometries of $L^p$

Epilogue

Bibliography

Index of Symbols

Subject Index