#### Top positive review

*4.0 out of 5 stars*Good, but probably not suitable to be used as primary text

Reviewed in the United States on March 20, 2019

This book is a Dover reprint of Shilov's "Elementary Real and Complex Analysis". Written in an old-school textbook style, it is not as conversational as some modern texts are, but it does offer numerous explanations here and there. The first nine (out of eleven) chapters are easy to follow. I can read them on train or in some other more distracting environments without difficulties and without needing pencil and paper. However, I am not sure if the book is suitable for novice learners, as its breadth and depth of topics seem somewhat imbalanced.

On one hand, some basic topics that can be found in many other introductory calculus or analysis texts are not covered here. This includes, most notably, connectedness of point sets, double sequences or series, and multivariate calculus. On the other hand, some topics that it does cover are pursued to probably too much depth. E.g. the chapter on metric spaces introduces not only completeness and (sequential) compactness, but also completion of metric space and precompactness. In the chapter on sequences and series, Riemann rearrangement theorem is proved, and guess what? A proof of Levy-Steintz theorem is also outlined across a whole page of exercises. This is strange, as the book was (if you have read the preface) supposedly a gentler introduction to analysis that physics or engineering students can follow.

Some symbols and terminologies in this book are old-fashioned. The intersection of two sets G and A is denoted by GA rather than G∩A. Cauchy sequences are called "fundamental sequences". "Compact" means sequentially compact rather than finite-cover compact. (Finite-cover compactness was referred to as "bicompact" in some older literature, but it is just called a "finite-covering property" in Shilov's text.) Some important results such as the intermediate value property of derivative or Lebesgue's criterion of Riemann integrability are relegated to exercises. An unsuspecting reader can easily miss them.

The greatest feature of this book is its bridging between high school and university mathematics. To my knowledge, Shilov's book is the only introductory analysis text that fully justifies that the exponential, logarithmic and circular functions (sine, cosine, etc.) defined in the modern way using power series are identical to the functions bearing the same names in high-school curricula. The whole tour is long. It begins with chapter 5 (Continuous Functions) and ends with chapter 8 (Higher Derivatives). Shilov proves that, if you want some functions called log, sin and cos to enjoy some familiar basic arithmetic properties that we learned in high school (without assuming continuity or differentiability), such as log(xy)=log(x)+log(y) and sin^2(x)+cos^2(x)=1 (and a few others), these functions must be real analytic and possess unique power series expansions. Conversely, functions defined by those power series do satisfy the properties we laid down before. Therefore the log, sin and cos defined by power series are really those we learned in high school.

I like how Shilov writes longer proofs. These proofs are often written in ways that are less distracting and highlight the key ideas better. For instance, in a typical proof of the uncountability of the reals using Cantor's diagonal argument, one needs to resolve the issue that the infinite decimal representation is not unique (e.g. 0.1000... = 0.0999...). However, in Shilov's proof, this becomes a non-issue because the diagonal is constructed not really by writing down new infinite decimals, but by picking new points in Cantor subdivision. While it is essentially the same diagonal argument, Shilov's presentation has one less technicality to worry about.

We may also compare Shilov's book with Rudin's "The Principles of Mathematical Analysis". In both books, there are proofs for the fundamental theorem of algebra, Riemann rearrangement theorem, Abel's theorem, L'Hospital rule and Taylor's formula. Rudin's proofs are often slicker and I can follow every step in his proofs, but I have troubles to pinpoint the gist of his arguments. To borrow Shilov's words, "the reader can only take off his hat in silent admiration". In contrast, I always find the central ideas of Shilov's proofs easier to grasp.

(That said, on the whole, I think Rudin organizes the topics better. But Rudin is probably even less suitable for novices than Shilov is, because it is too terse. )

Shilov has made several attempts to streamline the teaching of real analysis. For instance, in chapter 4 (Limits), he proposes a set-theoretic notion called "direction" and uses it to define the concept of limit. His new definition of limit is a genuine generalization of the traditional ones (such as the ε-δ definition), but more restrictive than the filters/nets in general convergence theory (cf. Dixmier's "General Topology"). Often in textbooks, the limit of a sequence, the limit of a function at a point and the limit of a function at infinity are formulated differently, but they are now unified. As a result, proofs about basic properties of limits are simplified and feel less repetitive. E.g. to prove that Riemann integrals are additive, one can just prove them for plain Riemann sums and pass them to the limit. Upper and lower Riemann sums are no longer required.

Unfortunately, these attempts are not thorough. So, his readers might not fully appreciate the merits of these attempts. Shilov's uses of "direction", for instance, are mostly confined to chapters 4 and 9. In other chapters, he basically resorts to the traditional definitions of limits. As another example, Shilov tries to introduce the concept of differential rigorously in section 7.3 (along the lines of Fréchet derivative, although this is not made explicit) without using any hand-waving "infinitesimals". Alas, in section 9.9 (Further Applications of Integration), the ghosts of infinitesimals reappear again.

The last three chapters of Shilov are not as good. While the presentation in chapter 9 (The Integral) is clear, the applications to areas and volumes are too restrictive. It is well-known that the general definition of surface area is a hard problem (cf. Schwarz lantern). However, as multivariate calculus is not covered, Shilov can only deal with areas of surfaces of revolution and volumes of solids with known cross-sectional areas in his book. He is unable to give a general definition of area even for a smooth surface. Having seen a general geometric definition of length of curve earlier in this chapter, this ad hoc treatment of area and volume is very unsatisfactory.

Chapter 10 is about complex analysis. Basic topics up to Laurent series are discussed. The exposition is a bit rushed. My opinions may change if I re-read this chapter for some more times, but at present, I don't recommend anyone to learn complex analysis from it. The proofs in chapters 10 and 11 are more convoluted than those in earlier chapters. A few of them have gaps. There are also occasions in which a proof assumes a stronger condition than stated in the theorem. Fortunately, the stronger condition (e.g. a function is integrable) is always met when Shilov applies the theorem (say, because the function concerned in the application is continuous), but such condition blunders are quite troubling.

The final chapter on improper integrals contains some results about the continuity or differentiability of parameter-dependent improper integrals that are rarely seen in other introductory calculus or analysis texts. The chapter ends nicely with a discussion of analytic continuation of Gamma function.

Throughout Shilov's text, there is a "hard analysis" flavour that I find quite intriguing, especially when most other introductory analysis texts written in English are slanted towards the soft analysis side.