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I am taking my first course in Real Analysis this fall. With my background in analysis limited, I knew it was going to be a difficult course, and I prepared all summer. However, with the online university situation, my real analysis professor has been insufferable and made the class so much harder. His lectures are 100% reading out loud his textbook, and he has banned all external conversation and materials. I have been doing well, but I am essentially just teaching myself analysis from a textbook I dislike and it is taking a real mental toll to adhere to deadlines on homework that is extremely difficult and is focused on finding counterexamples, rather than understanding the material.

My main and only goal is to pass a graduate real analysis qualifying exam. At this point, I think it would be easier to just teach myself everything, rather than stress and frustrate myself with this course in such an already stressful time.

Does anyone have any book or material resources that would be best for teaching oneself real analysis? I have been reading Royden's Real Analysis, and I find it very readable and good for an introduction. I still need other materials to get a variety of problem sets and information. Books specifically from a perspective of set theory and algebra would feel most natural to me. Unfortunately, I cannot go to a library to seek other books before buying them, so your suggestions would be very helpful.

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    $\begingroup$ One book I like very much is G H Hardy's a course of Pure Mathematics. It's old fashioned - originally published in the early 1900s - but written clearly with lots of examples and exercises. It maybe too dated for modern use but it is in print and you can find PDF versions on line. $\endgroup$
    – WA Don
    Commented Sep 30, 2020 at 10:03
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    $\begingroup$ Have you already read undergraduate real analysis books such as Abbot? For undergrad analysis I would consider Abbot and Analysis I & II by Terence Tao (and there are many threads with other suggestions). For measure theory, in addition to Royden I'd consider Zygmund and Wheeden, Folland, and Sheldon Axler's new measure theory book which looks quite readable. (And there are threads with other measure theory recommendations.) For manifolds and differential forms at advanced undergrad level, I'd consider Analysis on Manifolds by Munkres. $\endgroup$
    – littleO
    Commented Sep 30, 2020 at 10:05
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    $\begingroup$ @littleO I have not had any real analysis before, and I am mostly self-taught from Fitzpatrick, so I suppose I am looking for something closer to measure and Lebesgue theory $\endgroup$
    – user717258
    Commented Sep 30, 2020 at 10:08
  • $\begingroup$ A Primer of Real Functions by Ralph Boas, updated by Harold Boas, is fun to read and suitable for self-learning or for supplemental reading. $\endgroup$
    – awkward
    Commented Sep 30, 2020 at 12:06
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    $\begingroup$ Does this answer your question? Good book for self study of a First Course in Real Analysis $\endgroup$
    – user851668
    Commented Dec 23, 2020 at 2:52

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Understanging Analysis by S. Abbott is in my opinion a very clear and easy to follow book. Also, Elementary Analysis: The Theory of Calculus by Ross

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'Undergraduate Analysis' by Serge Lang is amazing. It is quite lucid and has a host of good problems too. He has a way of explaining things in such a way that it seems simple. I am not sure if I can say the book is from set theory or algebra perspective, but it does seem like it. However, it is quite expansive in terms of content and will require significant time to be devoted if you want to cover from start to end. But if real analysis is something you are going to be doing a lot in future, this book definitely covers most, if not all, the basics. To follow this up, there's 'Real Analysis' again by Lang.

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As Marcelo recommended, Understanding Analysis by Abbott is a great way to get started in real analysis, and if you're struggling to understand your lectures you should REALLY get a copy and start reading it ASAP. A few standard books:

Principles of Mathematical Analysis - Rudin: Very hard the first time you read it. But it's the gold standard for undergrad real analysis once you understand a bit of analysis. The first 7 chapters cover the standard curriculum in an extremely clean, lucid way. At some point during undergrad, you should work through this book. Exercises are good but a bit challenging. I felt like I learned more by complementing them with more straightforward exercises from Abbott first, and then attempting the exercises in Rudin.

Analysis I & II - Tao: Brilliant exposition and very accessible. Also written by one of the math world's rockstars. It's a bit unusual in its exposition though (spends a lot of time on the foundations, and doesn't really get to the analysis for a while). Worth getting a copy of these books.

Mathematical Analysis - Apostol: Didn't read much of it, but I've heard great things. Worth a look.

Real Mathematical Analysis - Pugh: Same as above. Another very good analysis book, but I'm not as familiar with it as I am with the others.

Overall, you really want to stick with the choice of topics from Rudin but read the analogous sections from Abbott first to get a little context. At this stage in your mathematical development, you really want to use several sources to learn the material from different perspectives. Use the "gold standard" book to determine the choice and ordering of topics, but look at some lighter introductions first to build the scaffolding in your mind first.

Stick with it! Every math major struggles with their first real analysis course. It's a right of passage, but you come out the other side with your "sea legs."

Some youtube lectures: https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL0E754696F72137EC https://www.youtube.com/watch?v=gJ1pYz1k0qM&list=PL7B37EFE678A682CE

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