Customer Reviews

Foundations of Analysis (Graduate Studies in Mathematic)

5 star:		(9)
4 star:		(1)
3 star:		(2)
2 star:		(0)
1 star:		(2)

 

 

Landau's most known book is this little masterpiece. If you want to see everything about numbers proved, from the beggining, assuming just logical and set-theoretical principles and the five Peano axioms, you will find it here. You will see the proof of why 1+1=2, for instance, or why a+b=b+a. Usually people learn analysis with a lot of pictures and assumptions, and every once in a while one asks himself: how does it all begin? Because sometimes you see something which ought to be evident proved, and something which ought to be proved assumed. I recall that when I first met this book I became amazed and read it through with a lot of willing. It is difficult reading, so be prepared. That's because Landau wanted to follow the axiomatic Euclidean style in its most pure way. So the book is in the non-merciful telegram style of presenting everything in terms of "Axioms", "Definitions", "Propositions". Few books before and after strove to reach such pure and clear presentation of arithmetic. Thank God some one had once the patience to write such careful and complete text! In this book the words of Edgar Allan Poe are more than anywhere true: "What I here propound is true:-therefore it cannot die:-or if by any means it be now trodden down so that it die, it will 'rise again to the Life Everlasting'".

The book is very good for people who want to be a high-school teacher of math, or be a mathematician. Even if you don't take a class with this book, read it on your own before taking real analysis. It will make your thought and logic complete and precise. A really nice training and practical preparation to do analysis.

The book is very simple and short. It deals with number system from natural to complex, gently. Simple things are usually not easy, though.

I took real analysis twice long time ago, but this book still improved my thinking of numbers very effectively.

I recommend this book to those who want to be precise and correct, no matter you are math or theoretical physics people.

And also for high-school students who want to know what pure mathematicians really do.

And also for independent thinkers of mathematical science, and would-be philosophers!

Ever wonder how many of those field axioms in the front of your calculus/analysis book were redundant and what were essential? Are you a skeptic, who is not sure that you can really derive the complex numbers from Peano's axioms? Are you someone who does not know what the hell Peano's axioms are, even though you do know what an Axiom is? Then this book is for YOU!

This book proves the properties of the complex numbers and some of their subsets, (positive integers, integers, rationals, and reals). The proofs are quite rigorous, and provide an excellent foundation for which to study calculus, real or complex analysis. I thought it was a wonderfull read, and surprisingly accessible, given its hieroglyphic initial appearance.

If you decide to read this book in several sittings, I recommend being sure you cover each of the small sections. If you can't finish a full section, restart the section. I think that if you do it that way the results will seem more cohesive.

I give this book one star, not because it's a bad book--of its kind,it is not--but to sound a warning and to counter some of the rave reviews here. In the preface to his famous calculus text, the great Richard Courant speaks of a "smug and presumptuous purism" which had crept into the mathematics texts of his time and place (Germany, '20s and '30s). I suspect he had Landau's "completely rigorous" (and in my opinion, completely unreadable) calculus text. "Foundations of Analysis" is cut from the same monk's-hairshirt cloth. IF you really like math books that consist almost 100% of "definition-axiom-theorem-proof; definition-axiom-theorem-proof" style writing, and that defy the reader to penetrate a completely closed system of elegantly presented and almost contemptuously concise pure truths (or at least, pure deductions from axioms), then Landau is definitely for you, or at least worth a serious try. But if you like math books that contain base, ignoble, impure elements such as motivation, intuitive insight, historical information about the development of the number concept, or just an occasional prose oasis, you're better off looking elsewhere (e.g. Feferman, Mendelson, Henkin-Smith-et al). By comparison to Landau, Hardy's "Pure Mathematics" seems chatty and chummy; it's also a much better book which gives more insight into the foundations of analysis in its first chapter than the whole of "Foundations". To my mind, Landau's book is, to quote a very old joke, not so much "rigor" as rigor mortis.

Most "textbooks" really consist of tutorial lectures: motivation, explanation, examples, as well as definitions, theorems, and proofs. But think of a novel: it does not consist (normally) of the author lecturing you about the themes of the novel, explaining what the novel is trying to do, etc. The novel just does it. It is a "text" to be read and discussed, to be considered, to be taught. Landau's Foundations of Analysis is just such a text, a mathematical one. The reader has to figure out much of what is going on and why. (Euclid is also such a text.) In other words you have to dig for yourself. The work of figuring it all out is part of the intended learning. It is not beyond a reader's individual power to do so. A teacher can help with the motivation, explanations, etc.; so can fellow readers. It is a good seminar book. Solitary readers will have to fend for themselves, and individual responses to the text not surprisingly vary. Readers with a mathematical background may find it an easy read; others may or may not find the challenge rewarding. It deserves its classic status and that so many admire it ought at least to make one curious about it.

Many have commented on the content. I will only add my admiration of the fact that from just a few assumptions one can create all of arithmetic and the basis for an easy proof of the intermediate value theorem, something I just couldn't imagine how to do when I first took calculus.

This excellent little book starts with Peano's Axioms and finishes with the complex numbers. It is a good source of challenging proofs to support a modern algebra course. As a graduate student I found this book to be a very helpful resource for my abstract algebra course. As the preface states, the material is easily accessible to the 7th grader, but a bit more challenging to others. Try it, you will enjoy it.

I understand the argumentation of the one star reviewer, BUT Richard Courant always said that in Mathematics there must be a balance and a COEXISTENCE between the purism and and the pragmatism. He said that in "What is Mathematics?" and in "Differential and Integral Calculus". "Foundations of Analysis" by Edmund Landau is a masterpiece of pure mathematics. What's wrong with that? "Methods of Mathematical Physics" by Richard Courant and David Hilbert is a masterpiece of applied mathematics, What's wrong wit that?

This book is very great indeed,if you are a mathematician or a mathematics enthusiast or a serious student of mathematics and also want to know why the common algebraic laws as commutative,associative and distributive holds or more concretely why 2+2=4 not 5 or 6 or 10000000000 then only you should go through this book.Prof. Edmund Landau has done the great job and the serious mathematics enthusiast will never feel the waste of money as he or she goes through the books and began explore great ideas and techniques the great mathematician offers to you and no ordinary teacher or professor can provide those master pieces.Only a master can provide those master strokes.

Dover, please put this classic back into print!!
This does for numbers what Suppes (1960) did for ZF set theory.

This little book is quite peculiar. It tries to give a detailed account of the natural, rational, real, and complex numbers, and the book tries to prove many of the small theorems that govern the arithmitic for these numbers. In all honesty, when I read this book a year ago I enjoyed it. However, now that I am more mathematically mature, I'm not too sure if I would enjoy it now. This book essentially uses induction ad nauseum and uses basic algebraic manipulation throughout the entire book. While these things arn't in themselves bad, anyone who is slightly mathematically mature (lets say at least one proof-based course) would easily be able to do many of these proofs in a mater of seconds. This is simply a result of the nature of the material that this book deals with. I'm not geniunely expecting to see complicated proof methods in here. However, these proofs offer little mathematical knowledge to the reader.
Another thing that I am confused with is the style that this book is written in. This book has less soul than Baby Rudin! While I am a fan of terse mathematical exposition, I feel that presenting this material in the brutal fashion that it does is a waste. I will not be using this book as a reference, so there is no point to have it in this terse style. Landau even said himself that this book is written for first year university students who are taking a calculus course. Maybe the students he was writing for are used to reading this style of math, but I do not feel that the theorem, proof, lemma style works very well here. Why present seemingly simple material in this terse style? It almost seems as if he wishes to make this material seem more difficult then it actually is. This book will only be used as a learning tool, and so I wish he had not written this book as if he was expecting it to be used as a reference. Not only that, I don't believe that a first year university student would gain much from having no motivation whatsoever.

When I had bought this book, I read many of the reviews and thought that this book would tell me everything I ever wanted to know about numbers. Sadly, it doesn't, and so here is some extra material that I feel would make this book worth having.

I wish this book gave detailed descriptions on how all of the numbers are constructed. He really only tells us how the reals are constructed via Dedekind Cuts. If he used the Von Neumann construction to make the natural numbers, equvilance classes to contruct the integers, etc... then I feel that this extra material would make this book a must-have (definitely 5 stars). However, this book takes for granted most of these, and as a result we are left with a handful of theorems that amount to proving trivialities. If you happen to find a book that includes this material, it would be a very good book to have!

While I can not truly reccomend this to anyone, what it does it does well, and I did learn a few things too (so three stars). However, I wish that I had found a more detailed book that went into more material than this one does. If you are considering buying this book, my advice to you is to find a different book that covers this material and more. It will be a much more rewarding experience. Unfortunately, I do not know of such a book. If you happen to come across one, please let me know!