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Apostol Analysis vs Shilov Analysis

  1. Jun 17, 2016 #1
    I will be taking analysis in the spring next semester. I am taking intro to proof, java programming, differential equations (did not get transfer credit), and either a probability course or a second semester linear, this semester. I feel I have a good understanding of differential equations at the undergraduate level thanks to Ross: Differential Equations. I can use my "free study time", learning some analysis.

    I have Bartle:Introduction to Real Analysis and Lay: Introduction to Analysis. I am going to most likely use Bartle and supplement with Lay. Anyways, after Bartle, should I use Apostol or Shilov analysis book? I can get both fairly cheap, for under 10 dollars each. But which one is considered a better book and will prepare for graduate studies in mathematics? Or do both books complement each other well?

    I also was gifted complete sets of Courant and Apostol, and Spivak by an instructor. Should I just diligently work through one of these and skip the baby analysis books, and go straight to Apostol or Shilov?

    I was leaning towards Shilov, because his Linear Algebra book is something I would like to work through. However, Apostol has his Calculus series and Number Theory Book. His number theory book was heavily recommended by two professors.
  2. jcsd
  3. Jun 18, 2016 #2
    This is very personal. You'll need to decide this for yourself. But I always kind of liked Apostol's analysis book, although there are plenty of better books than the ones you mentioned.
  4. Jun 18, 2016 #3
    Between the two, I'd go with Apostol. Realistically, I'd say use both, since I've rarely regretted adding a book to my library. There's almost always a different approach to proofs, explanations, or applications which I then like or find otherwise instructive. Reading the parallel sections in a supplementary text can often provide a key bit of insight into understanding what I'm studying in the main text I'm working through.

    For further down the line, you might want to consider Angus Taylor's General Theory of Functions and Integration to work through and use as a reference after your analysis course this fall. He covers measure theory from multiple angles, and the book is a great jumping off point for other areas of analysis. It's a wonderful "third text" in analysis, in my opinion.
  5. Jun 18, 2016 #4
    The book for the course is one authored by the professor teaching it. The book is called, "Introduction to Math Analysis by Lax." I go to a state school, due to financial reasons. My goal is to go onto graduate study for Mathematics. I could not find information on the book. However, I have a gut feeling that the book may not be rigorous enough to meet my goal. I was fond of Spivak, but I had to put it away because of my honors EM course was a tad difficult to me.

    Not sure if I would be wasting time using Spivak/Apostol/or Courant, then going to Bartle, then Shilov or Apostol analysis. I am quite new to proofs. Not sure if this makes a difference? i try to prove every theorem I have encountered so far. I will have about 8 months to prepare for this analysis course.

    Russian books are quite nice. I really liked the way Russians write books in general. I did like Gelfand, Tolstov, and Irodov. Not sure if Shilov's exposition is lucid like these other authors?

    What books would you recommend for someone in my situation?

    I know it is a personal preference, however, I do not think I have enough information to make an informed opinion on this matter. Sorry if I am annoying you.
  6. Jun 18, 2016 #5
    If you're new to proofs, working through Spivak from the very beginning would be an excellent idea.
  7. Jun 18, 2016 #6


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    I definitely agree to micromass. "Good book" means many different things to many people. The two books you mention ,Apostol's and Shilov's, are both good but I'd go with Apostol.

    As for the approach (intro first or straight to advanced), personally, I prefer to go with the more challenging option in order to spend my time effectively, provided that I have sufficiently good knowledge at the introductory level.. If I find some serious difficulties in my understanding of an advanced book, that cannot be resolved either by myself or by using the available resources, then that is an alarm that my introductory exposition to the subject is not sufficient enough. And this I think, is a reliable measure of what I really know and not what I think I do.
  8. Jun 18, 2016 #7
    I think I will go with Spivak. Spent a few hours on it today, and it is at right difficulty. Should take me 5 months to thoroughly go through it attempting every problem. Really user friendly and he motivates the material really well.

    I'll jump to Apostol once I am done, and reference Shilov. If I do not understand either book on a topic, then I can go to Bartle, then back to Apostol.
  9. Jun 18, 2016 #8
    I should give a disclaimer before I write anything else: I have not read either of the books (Apostol or Shilov) on analysis but I am currently working through Shilov's Linear Algebra. And since no one else here seems to have read any of Shilov's book (sorry if someone here has that I didn't know about, I just didn't see anything to indicate it in the responses), I'll go ahead and give my experience with his Linear Algebra book.

    His explanations are certainly as lucid as Tolstov, if not better in my opinion (I have read parts of Tolstov's Fourier Series if that was what you were referencing earlier). So if you like other Soviet-style textbooks (where explanations had to be kept short and lucid because of the shortage of paper - at least that is what I have heard with reference to Landau's books), then you will like his book.

    His proofs are clear enough that I believe one could work through them with only a basic understanding of the fundamentals of proofs, but still concise enough that one has to do a fair amount of thinking on one's own to really get the idea behind it. However, about half of the statements which he proves are not done Lemma-Proof-Corollary style and are instead done by a more inductive method (i.e.: "Suppose we have a matrix of order n"...many seemingly odd (at the time) manipulations then go here between the beginning and the theorem..."now we have found the statement known as Laplace's Theorem."). So depending on how you like to learn this style may be better or worse suited for you.

    With that said, you should know the difference between "if-then" and "if and only if" when written in a statement, have a rudimentary understanding of proofs by contradiction, and be used to the notations of set theory such as "∈","∩","∪", and "⊂" as well as proving if an element in a set is unique. But this is only from my experience with his linear algebra book, though I would expect his analysis book to have similar prerequisites.

    So, from my experience with his linear algebra book, I think that if you like other Soviet-style books then you will like Shilov. However, if you aren't comfortable with the inductive type style or the prerequisites that I listed above, then maybe another book might be good for you. Like I said previously, this is only judging from his linear algebra book, so his analysis book may be completely different.
  10. Jun 18, 2016 #9
    Whichever book you choose, make sure you start with
    How to Think About Analysis by Alcock
    I found this to be an invaluable resource before starting analysis.

    I like Lay from your list. FWIW, Here are the other analysis books that i like (not in any specific order)
    Schramm - https://www.amazon.com/gp/product/0486469131
    Estep - https://www.amazon.com/gp/product/0387954848
    Pugh - https://www.amazon.com/gp/product/0387952977
    Rosenlicht - https://www.amazon.com/gp/product/0486650383
    Krantz - https://www.amazon.com/gp/product/1584884835
    Thomson - https://www.amazon.com/gp/product/143484367X
    Ross - https://www.amazon.com/gp/product/1461462703
    Abott - https://www.amazon.com/gp/product/1493927116
    Carothers - https://www.amazon.com/Real-Analysis-N-L-Carothers/dp/0521497566

    However, my most favorite Analysis books are
    Analysis I and analysis 2 by Terence Tao
    Last edited by a moderator: May 8, 2017
  11. Jun 22, 2016 #10
    MidgetDwarf, I remembered from your old post, you stated that you liked Shilov's ERCA; I thought you already read it.

    Having read Shilov in details and Apostol for some chapters, I think Shilov is a great book to learn analysis rigorously. He also treats interesting topics such as directions when talking about the sequences and limits. As for his proofs, they are as concise as Rudin, but he gives enough explanations before introducing his theorems and proofs. I have to tell you that problems in Shilov are not challenging at all...

    Alternatively, I recommend Pugh's Real Mathematical Analysis. He treats topology very clear and intuitively, unlike Rudin and Apostol. If you are a fan of challenging exercises, Pugh is a way to go as his problems are all difficult (some from qualification exams too). How about you try Pugh, and maybe supplement it with Apostol or Shilov?

    As for those "transition" books like Spivak and Courant, I think it would be a better idea to just jump straight into analysis. Although I do not doubt that Spivak provides excellent introduction to the advanced calculus, I think it would be better idea to start with books like Rudin and Pugh and learn from grounds, and supplement with motivational books like Spivak.
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