Although the book finds its way to some sophisticated results, the main body
of each chapter consists of a lean and focused treatment of the core
... [Show More] topics
that make up the center of most courses in analysis. Fundamental results about
completeness, compactness, sequential and functional limits, continuity, uniform
convergence, differentiation, and integration are all incorporated.
What is specific here is where the emphasis is placed. In the chapter on integration,
for instance, the exposition revolves around deciphering the relationship
between continuity and the Riemann integral. Enough properties of the integral
are obtained to justify a proof of the Fundamental Theorem of Calculus, but
the theme of the chapter is the pursuit of a characterization of integrable functions
in terms of continuity. Whether or not Lebesgue’s measure-zero criterion
is treated, framing the material in this way is still valuable because it is the
questions that are important. Mathematics is not a static discipline. Students
should be aware of the historical reasons for the creation of the mathematics
they are learning and by extension realize that there is no last word on the
subject. In the case of integration, this point is made explicitly by including
some relatively modern developments on the generalized Riemann integral in
the additional topics of the last chapter.
The structure of the chapters has the following distinctive features.
Discussion Sections: Each chapter begins with the discussion of some motivating
examples and open questions. The tone in these discussions is intentionally
informal, and full use is made of familiar functions and results from
calculus. The idea is to freely explore the terrain, providing context for the
upcoming definitions and theorems. After these exploratory introductions, the
tone of the writing changes, and the treatment becomes rigorously tight but
still not overly formal. With the questions in place, the need for the ensuing
development of the material is well motivated and the payoff is in sight.
Project Sections: The penultimate section of each chapter (the final section is
a short epilogue) is written with the exercises incorporated into the exposition.
Proofs are outlined but not completed, and additional exercises are included to
elucidate the material being discussed. The sections are written as self-guided
tutorials, but they can also be the subject of lectures. I typically use them in
place of a final examination, and they work especially well as collaborative assignments
that can culminate in a class presentation. The body of each chapter
contains the necessary tools, so there is some satisfaction in letting the students
use their newly acquired skills to ferret out for themselves answers to questions
that have been driving the exposition.
Building a Course
Although this book was originally designed for a 12–14-week semester, it has
been used successfully in any number of formats including independent study.
The dependence of the sections follows the natural ordering, but there is some
flexibility as to what can be treated and omitted.
• The introductory discussions to each chapter can be the subject of lecture,
assigned as reading, omitted, or substituted with something preferable.
There are no theorems proved here that show up later in the text. I do
develop some important examples in these introductions (the Cantor set,
Dirichlet’s nowhere-continuous function) that probably need to find their
way into discussions at some point.
• Chapter 3, Basic Topology of R, is much longer than it needs to be. All
that is required by the ensuing chapters are fundamental results about
open and closed sets and a thorough understanding of sequential compactness.
The characterization of compactness using open covers as well
as the section on perfect and connected sets are included for their own intrinsic
interest. They are not, however, crucial to any future proofs. The
one exception to this is a presentation of the Intermediate Value Theorem
(IVT) as a special case of the preservation of connected sets by continuous
functions. To keep connectedness truly optional, I have included two
direct proofs of IVT based on completeness results from Chapter 1.
• All the project sections (1.6, 2.8, 3.5, 4.6, 5.4, 6.7, 7.6, 8.1–8.6) are optional
in the sense that no results in later chapters depend on material in these
sections. The six topics covered in Chapter 8 are also written in this
tutorial-style format, where the exercises make up a significant part of the
development. The only one of these sections that might benefit from a
lecture is the unit on Fourier series, which is a bit longer than the others. [Show Less]