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. He then develops the topology of metric spaces and single-variable analysis: sequences, continuity, differentiation, integration and sequences of functions. The book ends with multivariable analysis, differential forms and the Lebesgue integral. It is known as a concise, tightly argued treatment. Rudin wrote it in the early 1950s from his experience of teaching analysis at MIT, and he revised it 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. Metric-space topology is 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

This chapter covers finite, countable and uncountable sets, metric spaces, compact sets, perfect sets and connected sets. It 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

The chapter opens with convergent sequences, subsequences, Cauchy sequences and special sequences. It then turns to series, covering 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. It also 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. The book therefore reads as one argument rather than a collection of topics.
• A backbone for courses. The weekly schedules for the UW–Madison and UT Austin courses listed under "Common course assignments" below work through the chapters in order. These schedules make it easy to match a self-study plan to a course.
• 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. George M. Bergman at Berkeley has written supplemental material that is freely available online and helps with the many compact arguments:
  • "Supplements to the Exercises in Chapters 1–7…" (PDF);
  • a separate set of comments on the text (PDF);
  • both are collected in his undergraduate handouts index.

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. In a course, lectures and homework supply some of that scaffolding; self-studiers will need to supply more of it themselves, and Bergman's comments and exercises can help.
• 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. Complex analysis appears only in places such as Chapter 8.
• Later chapters outside most listed courses. Most listed courses concentrate on Chapters 1–7 or roughly Chapters 1–8; USNA's first-term calendar covers Chapters 1–4, while Purdue also covers most of Chapter 9. Chapters 10–11 are outside the stated coverage of all the listed courses.

Where It Fits in the Course

Rudin is the classical rigorous text, best read after a first proof-oriented course in analysis rather than as one. In the course syllabi below, it is the main text for courses whose stated coverage ranges from Chapters 1–4 in USNA's first term to Chapters 1–7 (or into Chapters 5 and 9) elsewhere.

Three other books in this list suit a student who finds Rudin too compressed and can serve as a gentler first pass or a source of motivation:

• 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.
• Understanding Analysis (Abbott) is an undergraduate text that motivates each topic before developing the theory.

One suggested strategy is to read the relevant section of one of these first and then return to Rudin's version.

Introduction to Real Analysis and Yet Another Introduction to Analysis are other analysis textbooks in the list, both 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: Read Chapters 1–7 in order, the span stated in several of the course syllabi listed below. At first reading, Chapter 1 can be skimmed for the field and order axioms. Chapters 2 and 3 are the foundation for everything else and deserve the most time.
• Single-variable analysis: In Chapters 4–8, work through the proofs line by line and write out the intermediate steps that Rudin leaves out.
• Second course: Chapters 9–11 cover functions of several variables, differential forms and Lebesgue theory.
• Alongside: Use a gentler text for motivation and Bergman's supplement for additional comments and exercises. The supplement covers Chapters 1–7, matching the stated coverage of the UT Austin and Berkeley courses listed below.

Common course assignments

Published syllabi show how courses divide the book. The dated entries below (USNA, Purdue, Ohio State, UW–Madison, UT Austin) all specify the third edition.

Recent (within about eight years):

• USNA SM331H (Fall 2025, honours): The detailed first-term calendar covers Chapters 1–4. (syllabus PDF)
• Purdue MA 50400 (Fall 2024): Chapters 1–4, 6–7, and most of Chapters 5 and 9. (syllabus PDF)
• Ohio State Math 5201 (Autumn 2019): Chapters 1–7 as the stated syllabus for the first semester of a two-semester sequence. (course page)

Older archived lists (still useful for pacing):

• UW–Madison Math 521 (Spring 2011): Material drawn from Chapters 1–8; the weekly schedule moves through Chapters 1–7 in order. (course page)
• UT Austin M365C (Fall 2011): "We will treat the material in chapters 1–7", with a weekly schedule covering Chapters 1–7. (course page)

Undated course pages / secondary descriptions:

• UC Berkeley Math 104 (Tataru): Rudin required; "Syllabus: Chapters 1–7 of the text." The page does not give a term year. (course page)
• Princeton MAT 215 (honours single-variable analysis): According to Principedia (a course-description site, not a syllabus PDF), the course follows roughly the first eight chapters. (Principedia)

For self-study, these schedules are a flexible reference: the dated recent courses range from Chapters 1–4 (USNA first honours term) through Chapters 1–7 (OSU) to Chapters 1–4/6–7 plus parts of 5 and 9 (Purdue). Chapters 9–11 beyond that core can wait.