Since sometime early this year, I have been self-studying mathematics. Slowly, I have been making somewhat noticeable progress. My ultimate goal is to begin exploring subjects Topology and Geometry, but I do not think that I have sufficient prerequisites. This summer, I have much more free time to fill these gaps and thus, I plan isolate myself for about two hours each day completing exercises and working through books, books covering the three main subjects I wish to learn this Summer, namely Analysis, Linear Algebra, Abstract Algebra, and Calculus of Several Variables (curves, surfaces, manifolds, etc). Books I've considered reading include:

  • For Analysis, Principles of Mathematical Analysis by Rudin , most likely supplemented by one of the following (my favorite so far being Tao):
    1. Analysis I - Tao
    2. Mathematical Analysis - Apostol
    3. Understanding Analysis - Abbott
  • For Linear Algebra, Linear Algebra Done Right by Axler. I read some of the first chapter and it looks awesome.
  • For Algebra, Elements of Abstract Algebra by Clark. I love the problem-oriented nature of the book, plus it is very concise, unlike Dummit and Foote's tome.
  • For Multivraiable Calculus and Differential Geometry, I have looked at
    1. Introduction to Topological Manifolds - Lee
    2. Calculus on Manifolds - Spivak
    3. Differential Geometry of Curves and Surfaces - do Carmo
    4. A Geometric Approach to Differential Forms - Bachman

In fact, I'd like to learn more, but five, possibly six books seems a little absurd. My basic goal is to get a solid undergraduate education myself this Summer. Mostly, by myself, that is. Luckily, I do have access to someone to check my work and solutions to exercises, so if I do not understand the theory and complete the exercises erroneously, at least I will know.

Moreover, is my ambitious self-study plan realistic? Is it humanly possible for an enthusiast with sufficient mathematical maturity and background in proofs, namely me, to power through six books in one Summer? In my personal opinion, although I have merely previewed most of them, the books above are thorough and readable (with the exception of Rudin; although thorough, it is not quite as readable as the other books, but for that reason I intend to supplement it with a more readable text, such as Tao). My only constraint is time - precisely, I am confined to three months. Considering that, how can I best allocate my time? Should I only do exercises from one book and read from another? Should I read two unrelated books at once or focus one subject at any particular time. With that, what is the most logical order in which to read the above books? Is there anything I could add to or remove from the above list to receive better a better foundation in undergraduate mathematics?

Thanks in advance.

Edit: Since asking, I’ve picked up Tao and Rudin’s analysis texts to study this summer. I appreciate all of the comments and advice I have received. Also, my schedule has changed so that I can dedicate much more time, hopefully 5-6 hours a day.

Edit 2: Due to further comments, I’ve decided to read Terry Tao’s book only.

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    $\begingroup$ Personally I think it's unrealistic. Mastering the books by Abbot and Axler would be a big accomplishment already. Why the rush? $\endgroup$ – Lorenzo May 19 '18 at 20:50
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    $\begingroup$ Perhaps one of these books at most, given 1-2 hours a day. You'll find that some of the concepts take a long time to grasp, and much much more to master! Normally this list takes a full 4-year undergrad course to learn. $\endgroup$ – adamG May 19 '18 at 20:59
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    $\begingroup$ Based on personal experience, it can take a full summer to learn decently well the material in Abbots analysis. You will make much more progress if you focus on one thing, and think about it critically, and really try to internalize its key ideas. $\endgroup$ – Lorenzo May 19 '18 at 21:04
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    $\begingroup$ just a comment, in my opinion there is a huge jump in difficulty from Abbot's book to Rudin's. $\endgroup$ – qbert May 19 '18 at 21:30
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    $\begingroup$ Also, I'm an ambitous student like you and I've learned that sometimes focusing deeply on less things is much better. Focus on understanding and you'll do well. Best of studies! $\endgroup$ – Leo Lerena May 27 '18 at 20:01

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