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Summer Self-Study?

  1. May 29, 2010 #1
    Summer Self-Study??

    Hi, I'm high school senior from Vancouver, Canada and I'm entering U of Chicago this fall! I've searched through the forum like literally hundred times and still unsure of what to do for this summer. So, to introduce myself, I took AP Calc BC in gr.10 but never pushed myself further into studying higher level mathematics. Because my school only offers up to IB Math 12 HL, I took the course in gr.11 which include basically AP Calc BC + vectors + matrices + statistics + etc and I self studied IB Further maths which is (Advanced) Geometry + Set, relations, and Groups + Series and Differential Equations + Number Theory + Graph Theory + (Advanced) Statistics&Probability. I'm fairly good at math I'd say; I made CMO and USAMO in grade 12 and without any intense training. And I'm also a fast learner. But it's been a while since I studied math seriously and I haven't also been properly introduced to formal proofs. I studied bits of multivariable calc and linear algebra but im not quite skilled at these areas. I really want to make into Honors Analysis at UoC this fall so I've searched around the forum to choose which to read but I just wanted to hear you guys' opinions.
    So here's the plan
    How to Prove It: A Structured Approach - Daniel Velleman (Currently Reading)
    Naive Set Theory - Paul Halmos
    Mathematical Analysis I/II - Vladimir A. Zorich
    Linear Algebra Done Right - Sheldon Axler

    I know it's a tight schedule and I will most likely not be able to finish all those anyways. But, I wanted to know if there are other alternatives or extra books that I should read to be prepared for Honors Analysis. I know of Spivak, Apostol, Courant. Only reason I chose Vladimir Zorich was because it seemed very comprehensive and ordered (but I don't exactly know its level). IMO, Spivak lacks content (not to say it's easy, I know it's the bible of rigorous single-variable calculus) and Apostol is somewhat dry and also seems sort of messy (+ it contains LA and DE as well as probability, areas which I'd like to study from other books) and Courant is supposed to be physically intuitive and has more to do with applications. but I don't think it would be that much of a jump from what I already know. Correct me if I'm wrong. Please comment on books by Zorich if anyone knows since it seems it's quite new and not many people know about it. Or else, recommend me other combination of books (or other books on Linear Algebra) that I should study off from. If you would still recommend Spivak, please recommend me a multivariable counter part to it!

    Thank you all in advance.
  2. jcsd
  3. May 29, 2010 #2
    Re: Summer Self-Study??

    I was never really all that impressed by How to Prove It. Learning the basic general techniques is not hard, and I always felt that the immediate next step is to pick up say an intro analysis text (e.g. Spivak) and just read it.

    Drop Courant and Apostol from the list. Courant does impart physical intuition, but Spivak is basically a modern version with a lot of good problems. Apostol's calculus text is dry, and I prefer Spivak's style of breaking up the text into smaller chunks (there are 30 chapters or so). Besides, Spivak IS the text used at UChicago anyways.

    As for the other texts, naive set theory is good to have around as a reference, but you won't need a lot of that. The most common method of transfinite induction used in honors analysis is probably Zorn's Lemma, and you can learn how to use it from various other sources (e.g. http://gowers.wordpress.com/2008/08/12/how-to-use-zorns-lemma/" [Broken]).

    Sure you can read Axler, but I don't really see the point. Having a good handle on matrix algebra and the computational aspects of linear algebra is more important. Reading wikipedia probably suffices (e.g. start at "Basis (vector space)" and see if the various terms are familiar). If you get around to Zorn's Lemma, try to use it to prove that every vector space has a basis. This is probably too far-reaching, but your long-term plan is to learn math right?

    I've skimmed Zorich before, and it looks very comprehensive. I can't say anything else about the text right now, but I could get back to you.

    I would make two other recommendations: Rudin or Apostol's analysis text. Rudin is very difficult to read, but if you can do it (and you should try say the first chapter), you won't need Spivak. Spivak will flesh out in detail some of the subtleties in undergraduate real analysis, but Rudin challenges you to find them on your own. Some of the more difficult exercises in Spivak are borrowed from Rudin anyways.

    Apostol's Mathematical Analysis is somewhere in between Spivak and Rudin in terms of difficulty. One of the main differences between Spivak and the other two texts is that Spivak refuses to discuss metric topology for the sake of adhering to more elementary exposition. But metric topology simplifies numerous results in analysis with only a bit more machinery, and the notion of a metric space is crucial in analysis (this will become evident in honors analysis). For this reason alone, it would be wise to go with Rudin or Apostol (Zorich likely introduces metric spaces). Having said that, Rudin and Apostol have a lot of the same theorems, but as I hinted at before, Rudin's proofs are very slick and very concise. Apostol will not only fill in details in a particular proof, but might also prove something related that could appear as an exercise in Rudin. This aspect makes Apostol a good choice for an intro analysis text.

    Finally, you need to buy Kolmogorov and Fomin's Introductory Real Analysis anyways (it's like $10 or something). If you do one thing this summer, read chapter 2 on metric spaces! It's not something you are expected to grasp right away without basic real analysis under your belt, but once you understand how to work with limits rigorously (i.e. epsilon-delta arguments), you can get a lot out of that chapter.
    Last edited by a moderator: May 4, 2017
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