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Mathematical Analysis I

Vladimir A. Zorich
Publisher: 
Springer
Publication Date: 
2016
Number of Pages: 
616
Format: 
Hardcover
Edition: 
2
Series: 
Universitext
Price: 
69.99
ISBN: 
9783662487907
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on
05/25/2016
]

A combined review of volumes I & II.


This is a thorough and easy-to-follow text for a beginning course in real analysis, at the sophomore or junior level. In coverage it is roughly comparable to Apostol’s Mathematical Analysis or Rudin’s Principles of Mathematical Analysis, and so is best thought of as a rigorous look at calculus. It is not an upper level or graduate “real analysis” book in the sense of Boas’s A Primer of Real Functions, or Stein & Shakarchi’s Real Analysis: Measure Theory, Integration, & Hilbert Spaces. There is for example no Lebesgue integral or measure theory, and there’s not a lot about convergence of series other than uniform convergence and point-wise and mean convergence for Fourier series. There is some modest treatment of topology and function spaces, but more as a point of view than as subjects in themselves. In coverage the book is slanted towards physics (mostly mechanics), and in particular there is a lot about line and surface integrals.

Weighing in at about 1300 pages, the present work is three times as long as Apostol and four times as long as Rudin. The extra length comes not from more topics or more depth, but because Zorich writes everything out in detail and because includes a large number of worked examples. It does cover some topics in greater-than-usual detail and does cover a few newer topics that are not in the classic works. This second English-language edition is a translation of the sixth Russian edition; that in turn is the same as the second Russian edition (published in 1997) except for corrections and the addition of 11 short appendices on various newer topics. The first English-language edition (published in 2003) was based on the fourth Russian edition and appears to be reproduced unchanged here; the difference is the addition of the appendices.

The exercises are challenging and there are a reasonable number of them (although generally fewer than the worked examples). They are nearly all to prove things and are results that appear in some other books as theorems in the body.

In the first volume the approach is very traditional, starting with the real line and functions of a single variable and then working up to \(\mathbb{R}^n\). Sets and functions are constructed first, and then the real numbers are developed axiomatically (Dedekind cuts and Cauchy sequences are introduced almost immediately, but completeness comes as an axiom and not a construction). Then limits and continuity are discussed, and there is a leisurely development of the derivative and the Riemann integral in single-variable calculus. The first volume rounds out with multi-variable differential calculus. There is a lengthy treatment of the implicit value theorem, including a lot of applications and a thorough description of constrained optimization.

The second volume takes a more advanced and much more general point of view but is also something of a hodge-podge. This is where we start to see a lot of modern analysis. The volume starts out with a fair amount of point-set topology leading to the contraction mapping theorem, then doubles back to reconsider multi-variable differential calculus in terms of differentials, i.e., linear mappings that represent the derivative at a point. Then it springs forward to multiple integrals, then back again to differential calculus, this time using differential forms and manifolds (this material is developed for general topological spaces, not just \(\mathbb{R}^n\)). There’s quite a lot on line and surface integrals and vector fields; this is again in the classical mold and has a strong influence from physics. Then we have four diverse chapters: on uniform convergence, functions defined by an integral with a parameter, Fourier series and transforms, and asymptotics.

There are some scattered topics that are more modern (I am thinking of things developed in the last hundred years). These include some convexity and inequalities, distributions, and a few modern examples such as the Haar basis from wavelets.

There are a few nonstandard or uncommon notations, but these will probably not be confusing once the student is used to them. For example, open intervals are written with reversed brackets, thus \(]0,1[\). The older forms \(\varliminf, \varlimsup\) are used for \(\liminf, \limsup\). The inverse hyperbolic functions are written using “ar” rather than “arg” or “arc”, thus artanh.

The index is extensive but spotty, and I had trouble finding things in it. For example, in volume I there is a nice exposition, ten pages long, of Lagrange multipliers, but this term is not in the index. Both the terms “Riemann integral” and “Integral, Riemann” are indexed, but they point to different pages.

Bottom line: Will be popular with students because of the detailed explanations and the worked examples. The book does seem like overkill to me, and I wonder how many curricula could make good use of it. It would probably take about two years to work through in a course, and at the end the students would be experts at all aspects of calculus, but I feel that most math students would be better served by moving more quickly in to advanced analysis topics. I do think the book is valuable as a reference, despite the weak index.


Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

See the table of contents in the publisher's webpage.