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I've recently been looking around the internet trying to find math texts to use for independent learning. With that in mind, I am hoping for books with some credibility in the math community (like a "standard" book, or credited well like Rudin's texts), but online reviews such as Amazon reviews have felt too anecdotal for me to feel like I'm buying a correct book for me. So, I was wondering if any of you could suggest books that you have heard about or learned from yourselves.

I'm a senior undergraduate math student, but it is not my plan to go to graduate school for math. I want to continue my studies, and so I'm compiling a list of books to buy and read based on topics that seem interesting/fundamental.

Some of these topics I have a background in, but having a resource and/or review couldn't hurt. And, I'm interested in the consensus of the community.

Here is my list of topics.

Fundamental:

  1. Set Theory
  2. Number Theory
  3. Real and Complex Analysis
  4. Finite and Infinite Vector Spaces (Linear Algebra and Functional Analysis I believe)
  5. Topology

Interesting:

  1. Combinatorics
  2. Graph Theory
  3. Geometry (Euclidian and maybe non-standard)
  4. Differential Geometry and Manifolds
  5. P-adic Number Systems
  6. Chaos Theory

Suggestions for more topics are welcome too.

To give context to my learning style, I've been learning Real Analysis from Steven Lay and Rudin, in Analysis With An Introduction to Proof, and Principles of Mathematics respectively, and I have no qualms with the theorem-proof style mixed with intermittent reflection and problems. I'm not sure if I would like a more conversational book, like what Paul Halmos seems to right, but I'm open to suggestions. My only issue is that sometimes I don't understand some points in Rudin's book; a few times paragraphs have felt ambiguous, but that could just be a fault of my own.

I know there have been many suggestions for books, but without sharing how I enjoy learning it's impossible to tell if those books work for me, hence this new post.

Future thanks.

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closed as off-topic by Chris Brooks, Rob Arthan, davidlowryduda Mar 26 '17 at 23:18

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Seeking personal advice. Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances." – davidlowryduda
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ you might find my recommended book list helpful: markjoshi.com/RecommendedBooks.html but it's more aimed at people wanting to do financial maths. Also my book "proof patterns" is intended to be generalist. $\endgroup$ – Mark Joshi Mar 26 '17 at 22:39
  • $\begingroup$ Any answer to your question is doomed to be based entirely on opinions both about the topics you should study and about the books you should study on a given topic. $\endgroup$ – Rob Arthan Mar 26 '17 at 22:52
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    $\begingroup$ You might find the math autodidact's guide or this old thread useful. However, I'm closing this question. $\endgroup$ – davidlowryduda Mar 26 '17 at 23:18
  • $\begingroup$ Okay, understandable. Thanks for the links. $\endgroup$ – Anthony Monterrosa Mar 27 '17 at 0:14

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