Elementary Real and Complex Analysis (Dover Books on Mathematics)

This book is not exactly five stars in reality. However, I feel obligated to give it five stars for its effect on my life and educational career. The book covers a lot of good content and does not overlap entirely with what is done in Rudin (from what I hear, the standard Analysis book in the USA). I feel it is better to start with Shilov before going to Rudin's book. The reason being is that after finishing Shilov's book (took a long time for me with self study), I read most of Rudin's book without working it all out with pen and paper. Shilov is a good place to start, but it has some issues that need to be stated before one begins it. I will reveal the issues with the book by telling the story of why I give it five stars... oddly enough.So, why was Shilov worth five stars to me?Let's start with a short story about me. Many years ago about when I bought this book, I was a physicist in training and I did not like taking someone's word for anything. In physics, we often proved things with mathematics I knew only how to 'DO', but had no idea how to prove that the math we were using was true or valid for the particular case in question (Just look at the Dirac delta). I did not like that situation, so I set out to prove all the mathematics that I used in physics (a journey I have now learned I will never finish, however, I still continue that pursuit). One of the first math books I started reading was this book.Shilov starts from an axiomatic approach to the real numbers, and proves the whole of basic algebra, sequences, sums, and single variable differential and integral calculus (plus, a lot of other stuff). The attempted level of rigor is slightly less than Rudin (which is way less than the Bourbaki collective's attempt, which itself has its own problems), however, you will find a few typos and, more interestingly, logical errors that are subtle, but glaring once realized. Now, there were only a few proofs that were not fully correct given the level of rigor, yet half of what I learned from Shilov was catching and correcting fallacious proofs. The other half that I learned from Shilov was the approach to proving something: you will learn it in your own way, but it's usually a clever trick, so look for a good trick that makes all the logic fall into place.Each half I learned from Shilov seems contradictory: if the guys giving bad proofs in the book, how can you be learning what a good proof looks like? With how rigorous most of the book is, I often wonder if the errors in the book are intentional to see if the student is cut out for it. That said, a book with imperfections can't receive a perfect score. However, I owe this book five stars, because it started me on my journey and it was perfect for what I needed to learn at the point in life I was at. Whether it can do the same for another beginner, I think it is likely. Just remember not every proof is correct in the book, but you'll figure out which ones are one way or another.

The level of this book seems to me to be almost perfect for advanced undergraduates and early graduate students of physics. Perhaps its most appealing aspects are its unified approach that covers both real and complex analysis together with its choice of topics: the absolute essentials.While there are many sections of this book that are very concrete and would be completely appropriate material even in an advanced calculus course, the overall approach to the theory is rather abstract, making its way to metric spaces very early. As such, the reader should have the background provided by  Schaum's Outline of Set Theory and Related Topics .In addition, I would not recommend this book as a first book on analysis. Sufficient preparation for this book can be obtained from works such as  Introduction to Real Analysis (Dover Books on Mathematics) .Chapter 1 provides an axiomatic look at the real numbers, while chapter 2 is more devoted to set theoretical matters and introduces the complex numbers.Chapter 3 dives into metric spaces, and chapter 4 provides a general formulation of limits in metric spaces based on the idea of a set-theoretical "direction". This approach was new to me.The material in these chapters is presented at a rather high level of abstraction, but the book turns much more concrete toward the end of chapter 4 where it provides quite a bit of material on numerical sequences.Chapter 5 continues in this vein, with an abstract first part on continuous functions followed by a concrete end with coverage of basic topics like the logarithmic, exponential and circular functions.Chapter 6 is on series both of numbers and of functions. It also covers power series in quite a bit of detail.Chapters 7 and 8 both cover derivatives while chapter 9 is devoted to integration, and even provides some very introductory material on elliptic integrals.Chapter 10 covers the core material on complex analytic functions, and chapter 11 ends the book with improper integrals. This is definitely the most difficult chapter in the book and pretty much everything that has been introduced up to this point is brought to bear here. It includes material on the gamma and beta functions, and some fussy limit arguments.There is no measure theory anywhere here, although there is a brief appendix on the topic that I thought was the weakest part of the book.While this content rundown makes the book sound very boring, it is actually anything but. Shilov's books are a pleasure to read, and he does a nice job of blending the abstract and the concrete together into a unified and appealing whole. The same can be said for his attempt to present elementary real and complex analysis in the same book.There are plenty of exercises, some easy, some rather challenging. Most have either answers or key hints in the back of the book.Many math students argue about the merits of this book versus Rudin. This is easily resolved: read them both. But start with this one.For physics student, though, the choice is much clearer: read Shilov. Then at some point in the future, if the need arises, take a crack at Rudin.

Adorei

buon acquisto per il prezzo ma la stampa è migliorabile - dopo un po', la piccolezza dei caratteri affatica gli occhi e la mente (lo spessore fa sì che la lettura sia appensantita dal lavoro per distinguere alcune lettere)

A superb book that is a pleasure to read, with extremely clear presentation, very thorough, and an enormous amount of material covered. There is ample originality here - for example the treatment of limits is refreshing and different to what one normally encounters- but you can also rely on the fact that while nearly 50 years old now, its material still seems extremely well chosen (there is little left out which one would consider elementary analysis, and he does not go off on specialised tangents).The book is not as easy to read for a complete beginner in analysis as some have suggested, but with some familiarity with basic concepts, you will get an awful lot out of it. I should say also that it does not shy away from more advanced concepts. Metric spaces appear in Chapter 2 without much fanfare and by the end of this chapter you find yourself more deeply immersed in topology than you may have expected. Limits and continuity are handled in a generalised way very quickly and what is said everywhere applies to n-dimensional spaces as well as the real line. Line integrals are given a brief but beautiful treatment with the use of the Stieltjes integral in Chapter 9 and complex analysis is in full throttle in Chapter 10 (though it receives brief treatments much earlier).The last three chapters are substantial, with chapter 9 giving a broad 100-page treatment of integration including techniques that are more at home in calculus books (rather than analysis). Chapter 11 continues this after the Complex Analysis Chapter (10).There are many typos which are probably not in the original (a translator would have spotted and corrected them, and judging from other reviews of Silverman's translations, I'm guessing he introduced them), but that is not a real impediment because the majority are obvious enough. In only two places did I waste over half an hour trying to figure out why a proof did not make sense only to conclude that a mistake (typo?) was the problem. The knowledge that there are such issues, however, can also make you suspicious of valid proofs where steps are skipped by Shilov (not many of these, but there are a few).Overall this is such a superb book that any negative comments must be relegated to minor nuisances for which one is amply compensated by the genius of the author, the supreme clarity of the writing and remarkably orderly presentation of concepts from the simplest to the most demanding.

As I've mentioned when reviewing another of Shilov's books, this book, too, is great in terms of the clarity and flow of the presentation - Shilov understands the needs of students. Consequently,. the subject is easy to study from this book