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Mathematical Analysis Fundamentals

A. E. Bashirov
Publication Date: 
Number of Pages: 
Elsevier Insights
[Reviewed by
Allen Stenger
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This book has ambitious goals. It is intended to be a first course in analysis and to be a transition-to-proofs course. It is also intended to be suitable for students whether or not they have already had calculus. (The author seems ambivalent on this last point, saying on p. xi that “This book aims to serve as a transition from calculus to rigorous analysis” but on p. xii that “Familiarity with calculus is not a prerequisite for this book.”

Overall I was not very happy with the book. The biggest reason is that it doesn’t flow well; it reads like a compendium of theorems (and proofs), gathered into chapters of related results, but having no real path or goals. For example, it states and proves the Baire Category Theorem, but does not use it for anything. It states and proves the Contraction Mapping Theorem, but only uses it once, to prove the existence of solutions of differential equations, and that usage is 120 pages later.

I also think the book overuses symbolism for students at this level. For example, Theorem 9.10 is the familiar result that every continuous function on a closed interval is Riemann integrable, but the entire theorem statement here is “\(C(a, b) \subseteq R(a, b)\).” I also think the book is weak on counterexamples; these are important for transition courses to show the pitfalls that rigor is supposed to protect us from.

I think it probably is possible to learn analysis from this book without first studying calculus, but the big drawback is that you wouldn’t know why any of the theorems are important or where they came from. The approach concentrates on abstract spaces such as metric spaces and function spaces rather than the real line and Euclidean spaces. For example, the entire discussion of continuity is in the context of metric spaces, with barely any mention of functions on the real line. The Intermediate Value Theorem is proved there, but as a consequence of connectedness, without explaining explicitly the importance of the completeness of the reals.

Very Bad Feature: No index. It’s difficult to use the book for reference, because it’s hard to find the theorems. There’s also no index of symbols, which in some cases makes it hard to decipher the theorems if you haven’t read the material leading up to them.

On the plus side, the book does have numerous good exercises that test the student’s grasp and cover some advanced topics, and it does give thorough coverage of much of classical analysis, including some of the more advanced areas that would never be touched on in calculus. It also has unconventional approaches to some theorems, which is refreshing in most cases. These approaches are alarming in a few cases, for example where the Lebesgue integral is defined in terms of the Henstock-Kurzweil integral and there is no measure theory (apart from a discussion of sets of measure zero in connection with Riemann integrability).

Another book with similar goals that does a better job of meeting them is Ross’s Elementary Analysis: The Theory of Calculus. The coverage is similar though less extensive. There are also many good introductory analysis books that do not attempt to also serve as a transition, such as Boas’s Primer of Real Functions.

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.

  1. Sets and Proofs
  2. Numbers
  3. Convergence
  4. Point Set Topology
  5. Continuity
  6. The Space
  7. Differentiation
  8. Bounded Variation
  9. Reimann Integration
  10. Generalizations of Reimann Integration
  11. Transcendental Functions
  12. Fourier Series and Integers