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What is an appropriate route to learn analysis?

  1. Feb 18, 2010 #1
    Can anyone give an outline of analysis with its subdivisions and the best books in that division?
    This is what I've come up with so far:
    1- Principles of mathematical analysis by.W.Rudin
    2- Real and complex analysis by.W.Rudin
    3- Functional analysis by.W.Rudin
  2. jcsd
  3. Feb 18, 2010 #2
    Rudin's books are difficult. Royden is easier and is a classic.
    A lot of the problems of analysis come from Complex Analysis and the theory of Partial Differential Equations.

    Learn complex analysis. Alfors is a classic but intensely rigorous.
    Take a PDE and learn about it. E.g. the heat equation.
  4. Feb 18, 2010 #3
    If you've never studied analysis before, you probably don't want to begin with Rudin's Principles of Analysis (which I hope you know is the most basic of his books). I find Rudin's proofs beautiful, but his proofs are extremely terse and his logic is often subtle. This is not a book where you can quickly read through the proofs. Be prepared to spend a lot of time filling in details on your own and understanding every word. If you read a sentence of one of his proofs and feel that you sort of understand it, chances are you haven't grasped the subtleties behind his argument. I prefer Apostol's Mathematical Analysis.

    As for Royden, I've only read the first 5 or 6 pages of his chapter on Lebesgue measure on the real line (chapter 3 I think?). He does a good job of filling in all the details in a proof, but given that measure theory is not exactly the beginnings of analysis, Royden should probably be saved for later. I think it's meant as a transition from undergraduate analysis to graduate analysis. I mean he does apparently touch upon the real number system, metric topology, and Riemann integration, but this is all to motivate more advanced analysis material rather than to simply basic analysis.
  5. Feb 18, 2010 #4
    the book by pfaffenberger/johnsonbaugh is another good one. it's similar to royden's in that they first do sequences, series, functions, limits, etc in R, and then do the same things again (& more) for general metric spaces.
  6. Feb 20, 2010 #5
    Thanks guys 4 the info. I took into consideration your suggestions but I saw a lot of praise for baby rudin so I find a used copy and i decided to go though it. Its hard but enjoyable and I recommend to everyone.
  7. Feb 20, 2010 #6
    You should try to see both sides of things, that is, both praises and criticisms, and consider the backgroud of these comments. As a first-course self-study material of mathematical analysis, I hold a negative point of view towards Rudin's series, since it is intended for those who have had solid knowledge of mathematical analysis and want to elevate himself to a higher level. Do not try to use it to establish some of your first important concepts in mathematics. If you have read Munkres' "Topology" before, you may understand what I mean by comparing it with Chapter 2 of baby Rudin. Another illustration is in page 21. After a long proof w.r.t. completeness, a word "isomorphic" will get you lost: we are working with a rather complicated mathematical object, that is, the Dedekind cut, how can we identify it with real numbers, that is, points of the real axis? Of course, the author here assume that you have already possessed some knowledge of abstract algebra. Have you? As the third example, the constraints imposed in definitions are often violated. This occurs frequently in the book when dealing with derivatives, especially in Chapter 10. Hard book does not mean it contains more things. This can be verified by taking a look at Apostol's "mathematical analysis".
    For self-studier, choosing a right book is more important than choosing a "good" book. Study of mathematics is a long-term project in your academic career, so don't expect reading a difficult book once and for ever and then grasping everything. Reading Rudin's series before one is well prepared, according to my experience, will most probably mean a waste of time and effort without any repay. So, please think twice before plunging into Rudin.
    Before study of analysis, one should have some prerequists, except for calculus:
    1)mathematical logic and axiomatic set theory
    For the latter, I recommend Karel Hrbacek, Thomas Jech "Introduction to Set Theory". For the former, I'm sorry I haven't found any suitable book for self-study.
    2)abstract algebra and advanced linear algebra
    3)general topology
    As I mentioned before, Munkres' "Topology" is the only choice.
    When self-studying analysis, the following books may be suitable and helpful:
    1)Mathematical analysis: Apostol's "mathematical analysis"
    2)Real analysis: Gerald B. Folland "Real Analysis: Modern Techniques and Their Applications". It is highly recommended to read Paul R. Halmo's "Measure Theory" beforehand.
    3)Complex analysis: Lars V. Ahlfors "Complex Analysis".
    4) Functional Analysis: John B. Conway "A Course in Functional Analysis".
    Any comments are welcomed.
    Last edited: Feb 20, 2010
  8. Feb 20, 2010 #7
    I agree with you that Rudin's books are not for beginners. In fact they are too hard for most mathematicians.

    I also agree that Complex Analysis is de rigeur though Alfors is a bit technical. But reading his book will also introduce point set topology and the idea of homotopy and homology.

    I think Functional Analysis is not for beginners.

    I also think it important to learn a PDE to see how analysis is used. Any PDE will do. The heat equation is good because it relates to Complex analysis.

    I see no reason to learn mathematical logic.
  9. Feb 20, 2010 #8
    axiomatic set theory by suppes & theory of sets by kamke are both good for self-study, although the one by kamke is narrower in scope & has no problems. for some reason i didn't get dictionary ordering until i saw it in kamke's book. that would good to know before doing munkres' topology text because (iirc) munkres uses the ordered square in a bunch of examples
    Last edited: Feb 20, 2010
  10. Feb 20, 2010 #9
    I highly second number 2 and 3 (haven't read the others). Folland is my favorite textbook. But I'm curious why you recommend reading Halmos' book before? Folland's book is actually used as the text for my measure theory course, and I find it a heck of a lot better for this than Royden (which I used previously).

    If you get Ahlfors (it's damn expensive unfortunately) make sure you get the one with the new proof for Cauchy's theorem. I'd only seen the one that relies on Green's theorem before (ugly proof, with partial derivatives under integrals and double integrals) and the new one with the converging nested rectangles was :!!)
  11. Feb 20, 2010 #10
    lol :rofl:
  12. Feb 21, 2010 #11
    Functional analysis is a departure from the previous two topics as it is more akin to infinite-dimensional linear algebra. You should have a solid grasp of abstract finite-dimensional linear algebra (ie., Axler & Lax) beforehand.
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