Understanding Analysis

Citation

Abbott, S. (2015). Understanding Analysis (2nd ed.). Springer. Undergraduate Texts in Mathematics.

Chapter Summary

Each chapter opens with an informal discussion section that raises a motivating question, develops a lean core of definitions and proofs, and ends with a project section, where the exercises carry the argument, and a short epilogue. Chapter 8 consists entirely of projects.

Chapter 1: The Real Numbers

The chapter opens with the irrationality of √2 as the reason a larger number system is needed, then gathers some preliminaries before stating the Axiom of Completeness and drawing out its consequences. It closes with cardinality and Cantor's Theorem, including the uncountability of the real numbers.

Chapter 2: Sequences and Series

The discussion shows that rearranging the alternating harmonic series changes its sum, a warning that finite arithmetic does not carry over to infinite sums. The chapter then develops limits of sequences, the Algebraic and Order Limit Theorems, the Monotone Convergence Theorem, subsequences and the Bolzano–Weierstrass Theorem, the Cauchy Criterion and properties of infinite series, with a project on double summations and products of series.

Chapter 3: Basic Topology of R

Starting from the Cantor set, which has zero length yet is uncountable, the chapter treats open and closed sets, compact sets, and perfect and connected sets, with Baire's Theorem as the project. The preface concedes that this chapter is longer than later chapters strictly need.

Chapter 4: Functional Limits and Continuity

Dirichlet's and Thomae's functions motivate a careful definition of a functional limit. The chapter covers continuous functions, continuity on compact sets and the Intermediate Value Theorem, and its project asks which sets can be the set of discontinuities of a function.

Chapter 5: The Derivative

The discussion asks whether derivatives must be continuous, using the family xⁿ sin(1/x) as test cases. The chapter proves that derivatives have the intermediate value property (Darboux's Theorem), develops the Mean Value Theorems, and ends with the construction of a continuous nowhere-differentiable function.

Chapter 6: Sequences and Series of Functions

New in this edition, the discussion recounts how Euler's manipulation of power series led to the sum of 1/n². The chapter treats uniform convergence (including its interaction with differentiation), series of functions, power series and Taylor series, which the second edition moved into the core of the chapter, and closes with a project on the Weierstrass Approximation Theorem.

Chapter 7: The Riemann Integral

The discussion asks how integration should be defined once it is separated from antidifferentiation. After the definition of the integral, integrating functions with discontinuities and the basic properties, the chapter proves the Fundamental Theorem of Calculus. The organising theme, according to the preface, is a characterisation of integrable functions through continuity, which the project completes with Lebesgue's criterion for Riemann integrability.

Chapter 8: Additional Topics

Six self-contained, tutorial-style sections: the generalized Riemann integral, metric spaces and the Baire Category Theorem, Euler's sum, inventing the factorial function (the gamma function, with improper integrals and differentiation under the integral sign), Fourier series, and a construction of R from Q.

Key Concepts

Chapters 1–2: The Real Numbers; Sequences and Series

• Axiom of Completeness: every nonempty set of real numbers that is bounded above has a least upper bound, the property that separates R from Q.
• Cardinality: comparing the sizes of infinite sets through one-to-one correspondences, leading to Cantor's Theorem.
• Bolzano–Weierstrass Theorem: every bounded sequence has a convergent subsequence.

Chapters 3–5: Topology, Continuity and the Derivative

• Compact set: a set in which every sequence has a subsequence converging to a point of the set; Chapter 4 studies continuous functions on such sets.
• Intermediate Value Theorem: a continuous function on an interval takes every value between any two of its values.
• Darboux's Theorem: every derivative has the intermediate value property, even when it is not continuous.

Chapters 6–7: Sequences and Series of Functions; The Riemann Integral

• Uniform convergence: convergence of functions at a single rate across the whole domain, the condition under which limits preserve continuity and, with extra hypotheses, differentiability.
• Taylor series: the power series built from a function's derivatives, together with the question of when it actually converges to the function.
• Fundamental Theorem of Calculus: the two-part statement relating integration and differentiation.
• Lebesgue's criterion: a bounded function is Riemann-integrable exactly when its points of discontinuity form a set of measure zero.

Chapter 8: Additional Topics

• Generalized Riemann integral: a modification of the Riemann integral, due to Kurzweil and Henstock, that integrates every derivative.
• Metric space: a set with a distance function, the setting for a more general Baire Category Theorem.

Critical Analysis

Strengths

• Questions before theorems. The preface states the aim of focusing on questions that give analysis its fascination: whether the Cantor set contains irrationals, whether derivatives are continuous or integrable, whether a smooth function equals its Taylor series. The opening pages of each chapter bear this out, and Scott Sciffer's review of the first edition in the Gazette of the Australian Mathematical Society singled out the discussion sections that motivate each chapter and the epilogues that point onward.
• Proofs written for beginners. The author says that brevity is deliberately sacrificed for detail, that most proofs come with discussion of strategy and relevant definitions, and that routine arguments are moved to the exercises. Reviewers reprinted by the publisher praise the exposition: D. Robbins in Choice (October 2015) called it extraordinarily lucid, and Mark Hunacek in MAA Reviews (June 2015) called the book the gold standard among single-variable undergraduate analysis texts.
• Projects and a revised exercise set. The second edition adds roughly 150 new exercises to about 200 retained from the first, including Moore-method exercises that ask students to test conjectures and look for examples that may not exist, plus three new projects (Euler's sum, the Weierstrass Approximation Theorem, the factorial function). Ryszard Pawlak's zbMATH review of the second edition singles out this didactic design as the book's distinctive feature.
• Easy to tailor. The preface marks every project section (1.6, 2.8, 3.5, 4.6, 5.4, 6.7, 7.6 and all of Chapter 8) as optional, notes that no later theorem depends on the discussion sections, and explains which parts of Chapter 3 the rest of the book needs.

Limitations

• Deliberately narrow scope. The book treats real-valued functions of one real variable only; the preface says complex numbers are never used and that the book makes no attempt to be a complete catalogue of results. Students who need multivariable analysis or a reference text will need another book.
• Integration stops short of Lebesgue. The Riemann integral is the main object; Lebesgue's criterion is a project, the Lebesgue integral is only outlined in an epilogue, and the generalized Riemann integral appears as a tutorial in Chapter 8.
• Metric spaces only as a project. General metric spaces appear in a single section of Chapter 8, so a course that wants topology in metric spaces from the start will find the core material confined to the real line.
• Little worked support for independent learners. The publisher lists the solutions manual as an instructor resource, and many key arguments in the project sections are deliberately left as exercises.

Where It Fits in the Course

Within this course list Abbott is a main text for a first rigorous course in single-variable analysis. Two gentler books prepare for it in different ways: Lara Alcock's How to Think About Analysis is described as a study guide on reading definitions and constructing proofs, and R. P. Burn's Numbers and Functions leads the reader through the same topics by sequenced questions. Either works well before or alongside Abbott for students who have not yet written many proofs.

For a more comprehensive and more formal treatment afterwards, Bartle and Sherbert's Introduction to Real Analysis and Walter Rudin's Principles of Mathematical Analysis are the natural next steps. Abbott's preface acknowledges Robert Bartle's books as an inspiration for the whole text and bases Section 8.1 on Bartle's article on the generalized Riemann integral. The preface also presents the book as preparation for a second course on complex-valued functions on more general spaces, the broader setting of more formal analysis texts; the epilogues and the metric-space section of Chapter 8 are a useful bridge to it.

How to Use This Book

The book is written for undergraduates meeting rigorous analysis for the first time; it appears in Springer's Undergraduate Texts in Mathematics series, aimed at third- and fourth-year students. The preface names a robust understanding of single-variable calculus as the only prerequisite and describes the proof-writing experience of a linear algebra course as valuable but not required.

A one-semester plan, following the preface's advice on building a course (designed for 12–14 weeks):

• Core: the main sections of Chapters 1–7, from the Axiom of Completeness to the Fundamental Theorem of Calculus.
• Trim if needed: in Chapter 3 the later chapters rely only on open and closed sets and sequential compactness; open-cover compactness and perfect and connected sets can be skipped, since direct proofs of the Intermediate Value Theorem are given.
• Discussion sections: read them even if they are not lectured; they prove no theorems needed later, but they introduce examples (the Cantor set, Dirichlet's function) that the rest of the book uses.
• Projects: choose one or two to work through in full; the author uses them in place of a final examination and as group assignments. Section 8.5 on Fourier series is the longest of the Chapter 8 topics.

For self-study, try to answer each discussion section's questions before reading on, and treat the exercises, including the Moore-method items, as the main test of understanding. Because solutions are intended for instructors, a study partner or a companion such as Alcock's guide helps with checking proofs. Students continuing to measure theory should read the Chapter 7 epilogue and Section 8.1 carefully.