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I started studying real analysis with Rudin's book, but it's really terse, i somehow made it through the first chapter (After reading 3 times :D) and solved some problems but here we are, i feel like i really didn't understand that chapter. So, i found two texts to study analysis and leave Rudin's PMA, they are :

  1. Amann and Escher's, 'Analysis' (3 volumes)
  2. Pugh's 'Real Mathematical Analysis'

Both are rigorous and intuitive. (Am i correct? This is what reviews says so?)

I liked Amann and Escher's analysis, but by reading reviews on Amazon and a answer on this platform, it says that they present the material in a generalized form and i am afraid that i miss the usual special case way and well i maybe at a disadvantage in that sense.

My Questions :

1)Should I just study Amann and Escher's analysis alone? 2) Or study from Pugh's RMA 3) Is reading Amann and Escher's analysis enough or am i going to miss some stuffs for a usual analysis course?

Mooonyyy
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    I learned analysis mainly from the books of Amann and Escher, and yes, the point of view is general, not particular. This is the reason that I liked these books. I don't know Pugh's books, however if youa re inetrested in a book more oriented to physics or applications probably you will like the books of Zorich. – Masacroso Mar 02 '23 at 14:27
  • @Masacroso Can you tell me a bit more about your experience through Amann and Escher's analysis? What about the difficulty level of problems/exercises? – Mooonyyy Mar 02 '23 at 14:51
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    I don't think that the exercises are hard, I did almost all of them in the first try. – Masacroso Mar 02 '23 at 15:03
  • @Masacroso Well, i guess that's fine. Um, i will do amann and Escher's analysis then. Any other suggestions? Oh yeah forgot to ask, you liked these books, right? Other than general point of view of the subject, other advantages? – Mooonyyy Mar 02 '23 at 15:11
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    just to say that the Amann and Escher books were my second books of analysis, my first one was the book Understanding analysis of Abbott, so I had a previous background. I don't know how good or bad are the books of Amann and Escher because I didn't read many other books of real analysis – Masacroso Mar 02 '23 at 16:10
  • @Masacroso As mentioned in my comment, in mr. Fra's answer, I will pair Amann and Escher's analysis with Stephen Abbott or some other introductory eal analysis text. Thanks for the help. – Mooonyyy Mar 02 '23 at 16:13

1 Answers1

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I think Abbott, Understanding Analysis, or Bartle, Sherbert, Introduction to Real Analysis are two modern classics that are often used instead of Rudin. Amann is great, but I think it might be a bit too much if it's the first time you see this material, especially if you would like something that explains the context a bit more than baby Rudin.

Fra
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  • Thanks for the suggestion. However, Bartle and sherbet doesn't teach topology until 11th chapter, Abott is nice but i guess i'd prefer a more rigorous book. – Mooonyyy Mar 02 '23 at 15:40
  • -.- I am a bit biased towards studying some topology after skimming through the textbook by Munkres, not a goal or something like just curious about metric spaces. – Mooonyyy Mar 02 '23 at 15:42
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    Your choice obviously, but in my opinion you would be better off studying topology, multivariate analysis and the Lebesgue integral somewhere else, and later, rather than trying to find a real analysis book that does everything, at least the first time around. Amann is the only one that does all of it well (and it takes 3 volumes, not 1), but, for example, you'll see him introducing (Riemann) integrals by using Banach spaces, which is nice, just probably not for a first course (and you will have to learn much more than is actually necessary to do some real analysis). – Fra Mar 02 '23 at 15:53
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    Believe me i am not trying to do everything in a single course. You are right that it will be better if i do things separately, but isn't Amann and Escher used as a first year textbook in Germany? I'd really love to do just 2 volumes of amann and Escher's analysis and pair it with some standard introductory real analysis (a lot of time to spare tbh) seems right? If no, i will just follow your suggestion. Thanks for the help! :3 – Mooonyyy Mar 02 '23 at 15:58
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    I think using a standard introductory book along side a Amann is a good plan, my main point is: make sure you stay grounded and get your hands dirty with tons of examples and exercises, inequalities, limits, series, etc... that's really the main skill a first course in real analysis is built for. – Fra Mar 02 '23 at 16:06
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    I will sure do! Thank you mr. Fra! – Mooonyyy Mar 02 '23 at 16:08
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    @Mooonyyy Abbott book is, as far as I remember, completely rigourous. But I agree in that its better to learn real analysis directly starting from point topology – Masacroso Mar 02 '23 at 16:12