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Analysis Good supplementary real analysis book

  1. Feb 2, 2016 #1
    So the course I'm taking doesn't have a textbook requirement just lecture notes as the study material. While these are sufficient I would like to supplement with an outside reference that is a bit more in depth / explanatory.

    It's your typical undergrad real analysis course covering:

    The least upper bound axiom, real numbers, compactness, sequences, continuity, uniform continuity, differentiation, Riemann integral etc.

    Please don't suggest Rudin etc. I lack the mathematical maturity for these, which I should be building in analysis... but I don't see how trying to use a book like this as a supplement to an intro uni course would do me much good. May work for other math geniuses, but doesn't for me.

    I had looked at Apostol's Mathematical Analysis. Seemed good covered all of the topics, but it's (from my point of view) a little dense. Covering topics in 1-2 pages and moving on. I don't mind dryness, and do in fact like his writing style / explanations being a fan of his calculus series. Would prefer a book with a little more explanation / discussion than statement as I find this helps me more than the latter.

    Any ideas?
  2. jcsd
  3. Feb 2, 2016 #2
    Your results may vary with this, but I bought a Dover book--Foundations of Mathematical Analysis by Richard Johnsonbaugh--and I've done some self studying with it. It seems to cover everything quite well, and I was able to jump into it after starting with Spivak. It's a Dover book, so you can get it for as cheap as 10$, if you're looking for something inexpensive.
  4. Feb 2, 2016 #3
    Just looked at the table of contents / some sections..... and wow. Literally a perfect match (more or less, about as close as I could hope) step by step to my course outline. Same style and cheap too. Perfect suggestion and supplement. Thank you for pointing me towards it, dover always has gems.
  5. Feb 3, 2016 #4
    Shilov and Sherbert.

    Im finding that Sherbet is more accessible, however Shilov is a better book.
  6. Feb 3, 2016 #5
    Real Numbers and Real Analysis - Bloch,
    Very Rigorous explanation about everything, it begins by the axioms of hoe the naturals exists, and go on by constructing the intergers with extremely clarity and rigor, PERFECT for self-study
  7. Feb 19, 2016 #6
    I'm not exactly sure you have the 'mathematical maturity' to tackle it, but I think Understanding Analysis from Stephen Abbott might just be what you're looking for. Of course it won't be easy, but it will build up your mathematical maturity. One feature I particularly like is that in some of the exercices, you have to generalize, prove or even find special cases of theorems you found in the preceding section. It really helps in getting the theory to stick.

  8. Feb 23, 2016 #7
    Shilov - Elementary Real and Complex Analysis
    Hoffman - Analysis in Euclidean Plane
    Tao - Analysis I-II

    I strongly recommend any of them as a first introduction to the analysis. Shilov is at the level of Rudin, but it has much more gentler explanation and clear remarks. Plus, it has quite interesting problems. Tao's books start with rigorous, detailed treatment of the number systems and set theory, and he gives very clever proofs. Hoffman is good if you have a decent background in the linear algebra; he treats the integration and differentiation very well.
  9. Mar 26, 2016 #8


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    I've seen Abbott recommended in a great many threads but the Bloch text was recommended on this site's guide for self-studying real analysis.
    Has anyone had a chance to compare them?

    Real Numbers and Real Analysis by Bloch (554 pgs)
    Understanding Analysis by Stephen Abbott (312 pgs)
  10. Mar 27, 2016 #9
    I'm working through Angus Taylor's General Theory of Functions and Integration right now. The entry requirements are about the same as baby Rudin, but it explains things in what, to me, is a more robust manner. The "exit level"of the text is much higher than baby Rudin, also. It also handles things that will be useful in further courses of study, such as measure on Rn and methods used in functional analysis.

    I wouldn't necessarily recommend it as your only text, but it's been a good reference already for me. Also, it's printed by Dover now, and the first edition is in many university libraries.
  11. Apr 12, 2016 #10
    Yes, I used both when I started out Real Analysis. I think they're both good books. They both have their good points: I liked how each chapter of 'Understanding Analysis' started out as a discussion around some kind of concept, and then moved on to more formal definitions. Real Numbers and Real Analysis, on the other hand, has special topics that aren't included in Abbott's (a great construction of the number systems starting from the natural numbers, a construction of the Weierstrass function if I recall correctly). The order is also different, a lot more similar to a Calculus course.
    In the end, I chose Terence Tao's Analysis I as my main text though, and I only use Bloch and Abbott as secondary texts. All three are good introduction to the subject of real analysis in my opinion. Hope I could help.

  12. Apr 13, 2016 #11
    I strongly recommend "Elementary Real and Complex Analysis" by G. Shilov. It is as rigorous as Rudin, but it has better explanations and motivations behind the concepts. It also integrates the discussion of metric topology, continuity, and sequence, rather than trying to discuss them separately as Rudin, which is a great for understanding. Plus, you get to enjoy a brilliant exposition from Russian mathematician.

    As the author states in the preface, the book can be read by anyone with low mathematical maturity.
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