Arthur Mattuck: Introduction to Analysis
Publisher: CreateSpace (Amazon) 2013, (previously published by
Pearson (Prentice-Hall div.), 1999)
Massachusetts Institute of Technology
The book was developed at MIT, mostly for students not in
mathematics having trouble with the usual real-analysis course.
It has been used at large state universities and small colleges,
as well as for independent study. Students evaluate it as
readable and helpful. The current printing, by CreateSpace and
at a reduced price, is the eighth, incorporating all known
significant corrections and a new Appendix F.
This book is meant for those who have studied one-variable
calculus (and maybe higher-level courses as well), generally
skipping the proofs in favor of learning the techniques and
solving problems. Now they are interested in learning to read
proofs, and to find and write up ther own: perhaps because they
will need this for the next steps in their chosen field, or for
intellectual satisfaction, or just out of curiosity.
There are two paths to this. Some books start with a great leap
forward, giving the definitions in n-space. This requires first an
excursion into point-set topology, whose proofs are unlike those
of the usual calculus courses and are a roadblock to many.
The path chosen by this book is to start like calculus does, in
1-space (i.e., on the line) and focus on the basic definitions
and ideas of one-variable calculus: limits, continuity,
derivatives, Riemann integrals, and a few more advanced
topics. It's done rigorously, but also in as familiar a way as
So from the start it will use as a source of examples what you
know (with occasional reminders): K-12 mathematics and basic
one-variable calculus, including the log, exp, and trig
functions. This takes up about two-thirds of the book, and might
be as far as you wish to go. It sounds like this is just
repeating calculus, but students say that it feels very different
and is not all that easy.
The rest of the book gets into techniques from advanced calculus
based on the notion of uniform convergence, and usually used
in lower-level courses without proof: differentiating infinite
series term-by-term, and differentiating integrals containing a parameter (the
Laplace transform, for instance). For the latter, it's finally
time to learn about point-set topology in the plane (i.e.,
2-space, but n-space is no harder). There's also for the curious
or needy an optional chapter with the most important facts about
point-sets of measure zero on the line, and a more powerful
integral: the Lebesgue integral.
Two appendices respectively provide needed and optional
background in elementary logic, and four more give interesting
applications and extensions of the book's theory.
(See below for the link to the Table of Contents for more details
about the topics and the order in which they are given.
Generally helpful features
--- Leisurely exposiion, with serious comments about proofs, other
possible arguments, writing advice; some semi-serious comments,
--- Attention paid to layout and typography, both for greater
readability, and to give readers models they can imitate;
--- Questions after most sections of a chapter to firm up what
you just read, with Answers of various sorts at the end of the
chapter: single words, hints, complete statements, formal
(See below for the link to Sample Text Pages to see examples.)
Mathematically helpful features
--- The language of limits is simplified by suppressing the N and
the delta when their explicit value is not needed in the
argument, replacing them with standard applied math symbols
meaning "for n large" and "for x sufficiently close to a". These
are introduced carefully and rigorously; some caution is needed,
which is described at the end of the Preface (see the link below
--- The book tries to go back to the roots of real analysis by
emphasizing estimation and approcimation, which use inequalities
rather than the equalities of calculus, but have a similar look,
so that many proofs are calculation-like "derivations" that seem
familiar. But inequalities require more thought than equalities;
they are often mishandled and warnings have to be given and repeated.
Looking at the book
For more details about what's written above, you can use
these links in order to get an idea of how it's written, and
what studying analysis from it will
Its five-page Preface gives some
idea of the book's origins and style, and toward the end more details about the
mathematically helpful features..
TABLE OF CONTENTS: pdf file
What's in the book (and by omission,
what's not), and
the order of topics.
SAMPLE SECTIONS: pdf file
These are a few sections, totalling about 15
pages in all, showing text material, Questions, Exercises, and Problems,
to give you a sample of the mathematical writing style and level. They are
selected from the first three chapters:
--- Chapter 1: Real Numbers and Monotone Sequences
--- Chapter 2: Estimations and Approximations
--- Chapter 3: The Limit of a Sequence
Information about earlier printings
PRINTING: There is only one edition so far, but several printings.
The printing is identified by a number sequence like
10 9 8 7 6 5 4
on the left-hand page facing the dedication page; the
sequence shown identifies the fourth printing, for example.
CORRECTIONS: The current inexpensive printing is the eighth; it
incorporates all the significant mathematical corrections needed
from earlier printings.
I would be grateful to hear about any further corrections needed.
as teacher or student;
write to: email@example.com
For those using printings earlier than the eighth, here are lists
of corrections. Bullets indicate the more
significant ones; none are major.
Mathematical corrections to the Third through the Seventh Printing
( pdf file )
Mathematical corrections to the Second Printing
(see also the corrections to printings 3-7 above)
( pdf file )
Mathematical corrections to the First Printing (see also
the corrections to printings 2 and 3-7 above)
( pdf file )
Arthur Mattuck's Homepage