Real Analysis

Halsey L. Royden’s Real Analysis has contributed to educating generations of mathematical analysis students. The 5th Edition of this classic text presents some important updates while presenting the measure theory, integration theory and elements of metric, topological, Hilbert and Banach spaces that a modern analyst should know. It assumes an undergraduate course on the fundamental concepts of analysis.

Part I continues to consider Lebesgue measure and integration for functions of a real variable. In this revision, the treatment of general measure and integration is moved to Part II rather than Part III; material formerly in Part II is placed in Part III and a brief Part IV. This brings measure and integration on Euclidean space closer to their origin, the case of real variables; it also presents the opportunity to foreshadow more strongly, in the context of general measure and integration, concepts which later appear in general spaces.

Table of Contents

I Lebesgue Integration for Functions of a Single Real Variable

Preliminaries on Sets, Mappings, and Relations

1 The Real Numbers: Sets, Sequences, and Functions

2 Lebesgue Measure

3 Lebesgue Measurable Functions

4 Lebesgue Integration

5 Lebesgue Integration: Further Topics

6 Differentiation and Integration

7 The Lp Spaces: Completeness and Approximation

8 The Lp Spaces: Duality, Weak Convergence, and Minimization

II Measure and Integration: General Theory

9 General Measure Spaces: Their Properties and Construction

10 Particular Measures

11 Integration over General Measure Spaces

12 General Lp Spaces: Completeness, Convolution, and Duality

III Abstract Spaces: Metric, Topological, Banach, and Hilbert Spaces

13 Metric Spaces: General Properties

14 Metric Spaces: Three Fundamental Theorems and Applications

15 Topological Spaces: General Properties

16 Topological Spaces: Three Fundamental Theorems

17 Continuous Linear Operators Between Banach Spaces

18 Duality for Normed Linear Spaces

19 Compactness Regained: The Weak Topology

20 Continuous Linear Operators on Hilbert Spaces

IV Measure and Topology: Invariant Measures

21 Measure and Topology

22 Invariant Measures

About the Author

Halsey Royden was born in Pheonix, Arizona. He earned a BA from Stanford University at the age of 19, and one year later, an MA. After earning a PhD from Harvard University, he returned to Stanford to join the Department of Mathematics, where he remained for his professional career. He spent several sabbaticals at the Institute for Advanced Studies, Princeton. Between 1973 and 1981, he was dean of the School of Humanities and Sciences. During 1973{1974, he was a Guggenheim Fellow. The rst edition of his Real Anal- ysis was published in 1964. His research interests were in complex analysis and dierential geometry.

Patrick Fitzpatrick was born in Youghal, Ireland. He studied at Rutgers University, where, at the age of 19 he earned an undergraduate degree and later a PhD. After post doctoral positions as a Courant Instructor at the Courant Institute, New York University, and an L. E. Dickson Instructor at the University of Chicago, he joined the Department of Mathematics at the University of Maryland, where he has remained for his professional career. He has spent several sabbaticals in Italy. During 1997{2007, he was chair of the Department of Mathematics. His book Advanced Calculus is published by the AMS. The rst coauthored edition of Real Analysis was published in 2010. His research interests are in topological methods in nonlinear analysis.