Customer Reviews

Analysis: With an Introduction to Proof (4th Edition)

5 star:		(6)
4 star:		(6)
3 star:		(1)
2 star:		(1)
1 star:		(0)

 

 

I bought this book because I have been looking for a Introductory analysis text that isn't too advanced, but yet doesn't gloss over the essential stuff, and I found it in Lay's book. For the self-studier, this book is excellent! I have several books on analysis: Shilov, Kolomogorov, Rosenlicht, Ross,etc... For the beginner, this book is superior to all of them. A plethora of examples. Also, a good range of problems:from straight forward problems requiring only the use of a definition to more advanced problems requiring a little thought. If you already have had some Analysis, then this book is probably not for you. But, if you are a student who wants to learn Analysis on your own, then this book would be hard to beat. After this book, one should be able to tackle "Papa Rudin". For according to Rudin, all that is needed to study his "Real and Complex Analysis" tome, is the first seven chapters of his "Principles of Mathematical Analysis". This book covers all that Rudin covers with the exception of Riemann-Stieltjes integration. On the whole, this is a great start! If proof-based math is new to you, then you will appreciate the first chapter on proofs. Would have given five stars, but I would have liked to seen Riemann-Stieltjes integration. That's really only nit picking, though.

I didn't think this book was going to be very good, but the author has "proved" me wrong ;-) This book starts out so basic that in my class (which was the first analysis course in our math department) we actually skipped the first 1/3 or so of the book. The first 9 or 10 sections consist of stuff like basic set theory, logic, definition of a function, etc. I would think that even the most elementary Analysis books would completely leave this out and expect that the reader is already familiar with this. So if you need it, this book will be a good resource for you.

Then the book goes into a very nice introduction to topology. Basic concepts like open/closed sets, accumulation points, compact sets, etc. Topology can be a little intimidating simply because it's _so_ abstract, but this book makes the basic concepts very easy to understand, and prepares one for a more advanced course in topology. Alot of (good) Elementary Analysis books leave topology out, but I'm glad this book contained it. It is a very interesting subject.

All the material in the book is explained probably about as easily as the concepts CAN be explained. If you still have trouble with it, you might consider a different major. Not to say that this book transforms a very difficult subject into a pathetically easy piece of cake because that's impossible, but the material is presented probably as easily as it can be in order to maintain precision and detail (which is the whole point of Analysis).

The book is definitely not running short in the examples or end-of-section problems department, so that is another plus. The problems at the end of each section range in difficulty from problems that almost exactly match an example worked in detail in the section, to fairly challenging problems. With enough time though the average student could probably do every problem at the end of every section.

I'd recommend this book for self study as well as a supplement to any introductory analysis course. If you have already have exposure to rigorous proof of calculus theorems, then this book will probably be too basic for you.

The reason this book got 4 stars instead of 5 is because of its utterly ridiculous price. Just as good is Elementary Analysis: The Theory of Calculus, ISBN: 038790459X, except that it doesn't include the section on Topology ...

This is fairly basic introduction to Principles of Analysis, on intermediate undergrad level, strictly in R^1. The only other similar book I'm familiar is Kirkwood. The books of Rudin, Apostol, etc present the subject on much higher level.

My original intention was to take a course with Rudin, but after I've realized I had a hard time digesting his style, I've decided to take more elementary course. I knew the course would be using Lay, so I got this textbook and tried to learn it on my own, but wasn't sure how I was doing and ended up taking the course (still with Lay) anyway. So I'm quite familiar with this textbook. The only topics we didn't cover is "series" and "sequences and series of functions".

Now overall I would say it's a mixed bag. First, the good things. The first few introductory sections on sets and proof techniques are excellent, highly recommended, that's how I learned how to prove. I found exercises very useful.

Now things I don't like. First, lots of typos. I think I had 4th edition, and still I've managed to find over 20 misprints, incorrect references, etc, etc, all were reported directly to author. Second, and that's probably more important, in several instances the proofs are too convoluted and not self-motivating. To be more specific, the proof of Heine-Borell theorem is less than adequate. It is correct, but that's the kind of proof you read and then entirely forget how it went. I remember on the first reading I didn't feel comfortable with this proof at all. When I discussed this book with professor I was going to take that course with, he (surprisingle) agreed with me and told me he would present a different proof (and he did, much better one). Another example: proof that the modified Dirichlet function is Riemann-integrable. The proof can be substantially simplified. In fact, I've managed to simplify it. Finally, the same professor told me Lay's presentation of Riemann integrals had some holes in them, so he used Kirkwood instead. In fact he told me he was making choice between Kirkwood and lay (but ended up choosing Lay because he didn't like Kirkwood's book layout. Kind of funny reason, I think.)

In any case, I think Kirkwood is a bit better for self-study. Unfortunately it doesn't have intro to proofs, logic and sets. Ideally you should have both books, if you plan for self-study.

(note: I did took the Principles of analysis, after I've finished that one with Lay, and did quite well.)

This is a very good book for someone to look at before going into an analysis class with Rudin. If you have never done proofs or seen metric spaces or uniform continuity, etc., this is a nice, but brief, intro. This book will NOT teach you analysis - you have to use Rudin for that. But it is great for acquainting/preparing you for Rudin.

Analysis at this level is probably the most challenging class for an undergraduate degree. However, this book made it very manageable. I found the introduction to proof very helpful. I encourage anyone who is using this book to study this chapter ahead of time. It will make the subsequent chapters a lot easier to handle. If it was not for this book and the outsdanting professor I had, I would never have passed this class. Go for it!

I have the third edition, which I purchased for self study after I ran into trouble in Kolmogorov and Fomin, Introductory Real Analysis, which I had purchased after I ran into trouble with the topology and real analysis assumed by O'Neill in Elementary Differential Geometry. The advantages of Lay's book are described very well in the editorial reviews above. The book is very clear in both layout and prose. The author anticipates questions and explains the reasons for strategems used in proofs. The logical connections among such concepts as open and closed, complete, compact, continuity, metric spaces, and topology are presented clearly. I am enjoying Lay's book and I anticipate that I will soon be resuming study in differential geometry.

amazing condition. easy to follow text and apply concepts. crystal clear directions, great for a student just starting off and with no previous knowledge of the subject.

This is a really good intro to Real Analysis. However, it's a bit pricey and not recommend for the autodidact because of this. If you need this book for a class, obviously you should buy it. It is a really good text to go along with a good lecture. In my class (junior level analysis) we hardly used it outside of taking some homework problems from it but it helped reading it throughout the semester.

However, if you just want to teach yourself mathematical analysis, there are better options. I'd recommend the following:

Definitely make sure you're acquainted with "regular" calculus (usually covered in 3 semesters) before you begin.

If you've never been exposed to proof, start with How to Prove It: A Structured Approach. Then read Introduction to Analysis (Dover Books on Mathematics). The latter is used at my university in the senior level analysis class. You'll get a lot more mileage out of these two books and at a fraction of the price.

(And for the autodidact, you should also read Principles of Mathematical Analysis, Third Edition afterwards. It's sometimes taught in a beginning graduate analysis class but it's usually taught in senior undergrad.)

Steven Lay is every bad math teacher you've ever had rolled into one giant ball of smarm and uselessness. Let's go down the checklist:

- does he make every single concept more complicated than it needs to be, giving you page after page of gobbledygook when he could sum it up in one or two lines? CHECK! (For instance, for a function to be injective, any horizontal line can cross the function's graph only once. For it to be surjective, there must be exactly one horizontal line that intersects the graph at every point. There; that's everything that Section 7 needs to say, but instead it goes on and on uselessly for 12 pages).

- does he quiz you on concepts and terms that he didn't both to define and/or explain? CHECK! (Actually, the book's layout is so poor that important terms and definitions are buried in the middle of paragraphs; often it takes 2 or 3 passes to find the important stuff. A decent math book will be designed for easy reference, but apparently Lay assumed that we'd all memorize his text the first time through.)

- does he take every opportunity to prove that he's smarter than you, and that he doesn't really want you to learn anything? CHECK! (Insult to injury: instead of providing answers to the odd-numbered exercises, he seems to have randomly chosen the exercises which are answered in the back of the book. Or more likely, he only chose the easier ones.)

- does every example he does rely on some obscure trick or property that won't be utilized in the questions at the end of the section? CHECK!

This book is bad enough on its own; it's even printed on cheap paper that rips if you look at it wrong. God help you if you get this book in conjunction with a lousy Analysis teacher. I give it 2 stars only because, at the very least, it doesn't weigh much and doesn't take up too much space in my backpack.

I have had three or four other books for the class this text is for and this one is my favorite. The author does a good job explaining the theorems.