Rudin(PoMA) , Herstein (Topics in Algebra)

[foxtrot_echo_]

I will be appearing for a Maths exam.
For which I will have to be acquainted with what's contained in Herstein's Topics in Algebra and Rudin's Principles of Mathematical Analysis.

I have got very little pure mathematics background. I have completed engineering so have 4 semesters experience of engineering mathematics.

I only know very basics of set theory (i.e that which is needed for digital logic).

I would like to know what book I should read before beginning these two books.

Can I begin them directly based on what I know?
If so, which should I start first Rudin or Herstein? or should I start both simultaneously.

P.S - I read first few pages of both books and found that it wasn't totally Greek to me. But I needed to read each page twice or thrice so that the material got through.

 

[qspeechc]

P.S - I read first few pages of both books and found that it wasn't totally Greek to me. But I needed to read each page twice or thrice so that the material got through.

This is common with math books. I don't think I have ever read a page of a math book in under one hour.

How much do you know about rigorous proofs? Boths books assume tacitly that you have some experience with proofs. For Rudin you should ideally have a course in rigorous calculus under your belt, e.g. from Spivak.

I suggest you start with a book like https://www.amazon.com/dp/0131481010/?tag=pfamazon01-20.

Herstein is not quite as difficult as Rudin in my opinion, so perhaps you should go: Lay, then Herstein, then Rudin. However, there is no reason why you can't read Herstein and Rudin at the same time (after Lay, of course). For the latter part of Rudin you will need to know some linear algebra however, which you should probably know from your engineering math, and is covered in Herstein (about chapter 4 or 5 or 6 I think?, I'm quoting from memory...).

 

[foxtrot_echo_]

How is Calculus by Spivak ?

I have it. Can I use it as a background instead of going through Lay's/Velleman. As I guess, calculus would be more famililar territory for me instead of methods of proofs ?

 

[qspeechc]

Spivak uses proofs, it is a calculus book that proves everything, so you will still need to know proofs to work through it. If you want to read Herstein and Rudin you will have to learn how to do proofs anyway. Some people say they have worked through Spivak without knowing how to do proofs before-hand; it depends on you if you can do it. Spivak would make a very good book to work through before Herstein and Rudin. I recommend you read it instead of Lay. If you are having trouble doing proofs in Spivak (you might not), then I suggest reading Velleman first.

Btw, I strongly recommend reading https://www.amazon.com/dp/144192941X/?tag=pfamazon01-20's analysis book in addition to Rudin.

 

[foxtrot_echo_]

ok. I will see if I can get hold of the proofs/analysis books.

I was asking if I can use Spivak directly is because, I haven't had much trouble understanding the proofs used by Spivak.

And anyways having done Engineering one has some notion of proofs. My point is , can formal study of proofs be skipped if one reads Spivak.

Anyways, thanks for the replies. I have more idea of what to do now than before.

 

[qspeechc]

If you can work through Spivak and do the exercises, then yes, you can skip formal proofs and go on to Rudin and Herstein. Here is how you should read Spivak: when you come to a theorem, cover up the proof in the book and try to prove it for yourself. If you can not do it yourself, look at the first line or two of the proof in the book then try again. I repeat: if you can work through Spivak and do most of the problems, then you do not need to read any other books before Hersein and Rudin. It is in fact probably the best book to read before Rudin and Herstein.

 

[foxtrot_echo_]

Thanks .

 

[mathwonk]

I agree with the advice you have been getting here. The point of using other books in addition to Herstein and Rudin, in my opinion is that those books do not really explain well the topics. Just compare Rudin's discussion of the fundamental theorem of calculus with that in books like Pugh, or Berberian's Fundamentals of real analysis. In the first place there is no section heading in Rudin even called "fundamental theorem of calculus" so it is harder even to find it. And when you do find it, even though he has supposedly treated lebesgue theory, he does not prove the lebesgue version of that theorem. He also defines lebesgue measure for R^n at the beginning, whereas Berberian treats just R^1 first, giving you a chance to understand the simplest case before going to several variables. Rudin has most of the material, but I have found it is hard for my students (and me) to learn from.

Herstein also is a book from which it is hard in my opinion to get an understanding of the topics. He somehow manages to present the proofs clearly without giving an insight into how they were thought of or why they work. He explains more than Rudin but again without real insight in my opinion. Herstein's problems were a valuable part of the book to me. I recommend M. Artin's Algebra as much better than Herstein, or as a good supplement. You might also compare some discussions in Dummitt and Foote. I have criticized certain aspects of that book, but it has many clear explanations and a huge set of useful problems.

The point is, Herstein and Rudin are standard references, but I think many other books offer more insight and better explanations. In particular, anything written by Michael Spivak, Michael Artin, or Sterling Berberian, is always clear and helpful.

I must admit professional analysts I know like Rudin, why I am not sure. They seem to admire the elegance (read brevity), which is the opposite of what most students want. Even strong professional algebraic geometers I know are not nuts about Herstein, althiough some students who just like to memorize proofs like it. Also some students like me remember the fun of working his problems. But these problems are themselves like little puzzles, fun but not too important or useful.

 

[mathwonk]

Note the helpful remark on p. 375 of Pugh contrasting Lebesgue's original definition of a measurable set with that of Caratheodory. You will almost never find anything like that in Rudin. Both Pugh and Rudin use the same somewhat almost tautological treatment of differential forms. At least Pugh admits this and tries to explain why. A real understanding of differential forms requires an honest treatment as in Spivak's Calculus on manifolds, or the book of H. Cartan. They are patiently explained with many examples as a sort of "volume measure" in the book of David Bachmann, A Geometric Approach to Differential Forms. This book was read as a project here on PF some years back. Basically they are an abstract version of determinants, which must play a role in parametrized integration because of the change of variables formula.

 

[foxtrot_echo_]

@mathwonk Yes , when I read a few pages of Herstein and Rudin , I did experience what you just told.
Apart from the difficulty level , I didn't get what the motivation was behind certain things.
I.e I found the brevity aspect going against a beginner such as me.

So I have decided to start with Calculus - M Spivak.

I couldn't get hold of Calculus on Manifolds or the other books you mentioned.But will surely try to do so .


Thanks again for all the help.

 

[foxtrot_echo_]

which is easier Pugh or Berberian? After I finish Spivak that is.

 

[qspeechc]

Don't attempt too much at once. Usually I had one or two books for a course. Trying to read too many books at once on one subject is distracting. The books have different contents, in different orders, presentation, emphasis, terminology, level of presentation, etc., so I would not recommend trying too many books at one time.

Pugh will be closer in content to Rudin than Berberian (I haven't read the latter though). I would suggest reading Rudin and Pugh together, and then afterwards, if you are interested and want to read more, then you can look at the other books mathwonk suggested.

The only algebra book I have read is Herstein, but mathwonk says Artin is better, so I would suggest you read Herstein with Artin.

 

[foxtrot_echo_]

No I am not attempting too much. hehe. After reading a few pages of Rudin and Herstein I have kept them out of sight currently. I am doing Spivak - Calculus.

I just asked about the future books in view of making arrangements for them - i.e borrowing from someone.

Only after Spivak will I attempt those other books. I have read a few chapters from Spivak and I found the book quite helpful.Let's hope it serves as a good start and that I can successfully finish it in 2-3 months.

And again thanks qspeechc and mathwonk,your comments have been both helpful and encouraging.

 

[mathwonk]

herstein is very seductive. it is good in some ways, but somehow after learning from that book, one comes away with no understanding of the material. I used to say it goes in one ear and out the other. artin is harder to read than herstein but offers more insight. of the many books on my shelf, herstein is one i almost never refer to for anything. also rudin. rudin is a book i can only see something useful in after i have understood it from some other source. berberian's books are not as popular, so are available for a pittance on the web. i recently bought berberian's analysis book for a small sum at amazon used books...
well amazon wanted $71.

here's one for $10.

Stock Image
Fundamentals of Real Analysis (Universitext) (ISBN: 0387984801 / 0-387-98480-1)
Berberian, Sterling K.
Bookseller: Maximumex Books
(Rancho Dominguez, CA, U.S.A.)