Welcome to Real Analysis

In Welcome to Real Analysis, Benjamin B. Kennedy offers a complete and self-contained introduction to analysis that covers, and moves beyond, the foundational work of single-variable Calculus. Its subtitle, Continuity and Calculus, Distance and Dynamics reveals the author’s ambitions to explore topics across calculus and discrete dynamical systems. Kennedy’s aim is to ensure that a student’s first exposure to Real Analysis simultaneously enhances their understanding of first-semester calculus, and also shows how core ideas from calculus (most notably convergence and continuity) can be used in flexible and diverse ways. Many of the decisions made throughout the text about content focus are in service to this aim.

 

In terms of content: this text offers a short introductory chapter on the essentials, before launching into the main matter of the text. In keeping with its goal to remain maximally flexible, the text works from the viewpoint of metric spaces (introduced and focused on in Chapter 2, where topological properties of metric spaces are also introduced). Although most of the initial focus is on Euclidean space metrics, other metric spaces (such as sequence spaces and spaces of continuous functions on compact sets) are introduced up at appropriate moments. The text proceeds by introducing the two ``core ideas’’ from calculus: a chapter is devoted to sequences and sequence convergence, followed by another chapter devoted to properties of continuous functions from one metric space to another. The text continues with three chapters that are core elements of building the theory of calculus: compact sets (from the viewpoint of (sub)sequential compactness), derivatives, and integrals. Finally, the text concludes with three chapters that are supplementary but not less interesting: one on sequences of functions; one on chaos; and one on fractals, introduced via the Hausdorff metric.

 

A triumph of this book is the complete integration of dynamics from an early moment in the text. Discrete dynamical systems are introduced near the start in chapter 3 (immediately following an early discussion of sequence convergence, and before a number of “classic” results such as the algebraic limit theorems). This early adoption of dynamics allows for many topics – such as contraction mapping theorems and the existence/uniqueness of solutions to IVPs – to be explored as an integral part of the text, rather than remaining part of an auxiliary chapter. In this approach, I think Kennedy is extremely successful.

 

In terms of writing: this book is clear, challenging, and rewarding in its exposition. The ordering of topics is well-vetted; examples and proof details are well-chosen; and the exposition is well-written and imminently readable. The text offers reading questions after each section, as well as a variety of exercises at the end of each chapter. I appreciated both forms of questions; the “exercises” are varied and completely appropriate, while the “reading questions” allow for students to check their comprehension (and model the kinds of questions we want students to ask themselves as they read any mathematical text). Additionally, the text benefits from “analytical advice” text boxes that offer just-in-time discussions of proof styles and big ideas in proof-writing. 

 

However, as a small word of warning: I think students would struggle with this text if they do not have (1) a solid foundation in proof-writing, (2) a certain level of comfort in manipulating algebraic equations and inequalities, and (3) a willingness to read the textbook deeply and carefully. Chapter 1: Essential Tools flies by in a hurry. It introduces students to each of the major building blocks of the class, but does not dwell on any of them for very long. This is exemplified in an early section on inequalities, which introduces the Triangle inequality (using both absolute values and norms), as well as the Schwarz inequality. The text offers proofs of these results in an abstract and largely context-free manner, and draws on them throughout the rest of the text. However, at first pass, these feel complicated and largely unmotivated, and would be challenging for a novice proof-writer to follow.

 

In summary: This seems to be an excellent text which truly highlights how the core elements of analysis can be applied in a number of powerful ways beyond calculus. I would argue against using this text in a course that uses Real Analysis as an “introduction” to proof-writing, as such students might find this text a bit of a struggle. However, a well-prepared undergraduate will benefit immensely from the exposition here, and I would recommend the text to such a student without reservation.

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John Ross is an assistant professor of mathematics at Southwestern University.