Principles of Mathematical Analysis

Citation

Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill.

Chapter Summary

The third edition has eleven chapters. Rudin first builds the real and complex numbers, then develops the topology of metric spaces, then single-variable analysis (sequences, continuity, differentiation, integration, sequences of functions), and finally multivariable analysis, differential forms and the Lebesgue integral. The book is known as a concise, tightly argued treatment; it was written in the early 1950s from the author's experience of teaching analysis at MIT and revised in 1964 and 1976.

Context and Argument

Rudin has said his aim was to present a beautiful area of mathematics in a well-organised readable way, concisely and efficiently, with complete and correct proofs. The book treats classical real analysis in a modern, axiomatic style, using metric-space topology as the foundation for limits, continuity and compactness.

Chapter 1: The Real and Complex Number Systems

Ordered sets, fields and the least-upper-bound property are used to construct and characterise the real and complex numbers. In the third edition the construction of the reals by Dedekind cuts has been moved to an appendix.

Chapter 2: Basic Topology

Finite, countable and uncountable sets, metric spaces, compact sets, perfect sets and connected sets. The chapter treats the real line as a metric space and supplies the topological vocabulary used in the rest of the book.

Chapter 3: Numerical Sequences and Series

Convergent sequences, subsequences, Cauchy sequences and special sequences, followed by series: tests such as the root and ratio tests, and absolute versus conditional convergence.

Chapter 4: Continuity

Limits of functions and continuity, the types of discontinuity, and properties of continuous functions on compact and connected sets.

Chapter 5: Differentiation

The derivative, the mean value theorems, L'Hôpital's rule and higher derivatives.

Chapter 6: The Riemann-Stieltjes Integral

The Riemann-Stieltjes integral is defined and its properties established, including its use on rectifiable curves.

Chapter 7: Sequences and Series of Functions

Uniform convergence and its consequences for continuity, integration and differentiation, followed by equicontinuity and the Stone-Weierstrass theorem.

Chapter 8: Some Special Functions

Power series, the exponential and logarithmic functions and the trigonometric functions, with an extension to complex functions. The chapter also takes in the fundamental theorem of algebra and Fourier series.

Chapter 9: Functions of Several Variables

Linear transformations and differentiation in several variables, including the inverse and implicit function theorems, and integration in higher dimensions.

Chapter 10: Integration of Differential Forms

Differential forms, partitions of unity and Stokes' theorem, which generalises several classical theorems of vector calculus.

Chapter 11: The Lebesgue Theory

Measure and the Lebesgue integral, measurable functions, and the space of square-integrable functions.

Key Concepts

Chapters 1–4: Foundations

• Least upper bound property and ordered fields: the property that characterises the reals (Chapter 1).
• Metric spaces, compact and connected sets: the topological framework (Chapter 2).
• Cauchy sequences and convergence tests: completeness and series (Chapter 3).
• Continuity and its relation to compactness (Chapter 4).

Chapters 5–8: Single-variable analysis

• Mean value theorems (Chapter 5).
• Riemann-Stieltjes integral (Chapter 6).
• Uniform convergence and the Stone-Weierstrass theorem (Chapter 7).
• Exponential, logarithmic and trigonometric functions (Chapter 8).

Chapters 9–11: Several variables and Lebesgue theory

• Inverse and implicit function theorems (Chapter 9).
• Differential forms and Stokes' theorem (Chapter 10).
• Lebesgue measure and integration, and L^2 (Chapter 11).

Critical Analysis

Strengths

• Economy and rigour. The proofs are complete, concise and carefully ordered. The MAA lists the book in its Basic Library List of recommended undergraduate library titles, and it earned Rudin the Leroy P. Steele Prize for Mathematical Exposition in 1993.
• A coherent path from numbers to Lebesgue integration. The topology of Chapter 2 is used repeatedly in Chapters 3 to 7, and the same framework carries into the multivariable chapters, so the book reads as one argument rather than a collection of topics.
• Reach. Beyond a first course, the book includes differential forms with Stokes' theorem and an introduction to Lebesgue theory, so the later chapters can serve a second analysis course.
• Support material exists. Supplemental comments and exercises for Chapters 1–7, written by George Bergman at Berkeley, are available online and help with the many compact arguments.

Limitations

• Terse for a first encounter. The style is dense and gives little motivation, so beginners and self-studiers are likely to need a gentler companion.
• Theory only. The book is about the theory of analysis. Readers who want examples from physics, economics or computing must find them elsewhere.
• Lebesgue theory in a single chapter. The final chapter is an introduction rather than a full course on measure theory, and complex analysis appears only in places such as Chapter 8.

Where It Fits in the Course

Rudin is the classical rigorous text, to be read after a first proof-oriented course in analysis rather than as one. How to Think About Analysis is a study guide on how to read definitions and build proofs, Numbers and Functions: Steps into Analysis is a question-driven introduction, and Understanding Analysis is an undergraduate text that motivates each topic before developing the theory. Any of these suits a student who finds Rudin too compressed.

Introduction to Real Analysis and Yet Another Introduction to Analysis are other analysis textbooks in the list; their descriptions place them as rigorous introductions to the same core material. Within this list Rudin is the book that carries the core material furthest, through differential forms and Lebesgue theory.

How to Use This Book

The book is for mathematics students with a solid calculus background and some experience reading proofs, for example in an honours course or at the start of graduate study. It is not an easy first read, but it rewards slow work with a pencil.

A practical plan:

• First pass: Chapters 1–7 in order. Chapter 1 can be skimmed for the field and order axioms at first reading; Chapters 2 and 3 are the foundation for everything else and deserve the most time.
• Single-variable analysis: Chapters 4–8, working through the proofs line by line and writing the intermediate steps that Rudin leaves out.
• Second course: Chapters 9–11 for functions of several variables, differential forms and Lebesgue theory.
• Alongside: a gentler text for motivation, and Bergman's supplement for additional comments and exercises.