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Mr.Miyagi

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**What to read after "Understanding Analysis"**

I have worked through Abbot's "Understanding Analysis" thoroughly and would like to learn more about the subject. My goal is to gain a good understanding of real and complex analysis and I also want to work my way up to differential geometry.

I think I have had an average course-load of mathematics for a 3rd year student of physics. Most important to my question, linear algebra (Leon), calculus (Stewart) up till real analysis (Abbot) including vector calculus (Colley). Also the basics of complex analysis (Saff and Snider) and some tidbits here and there as needed for the physics.

Since my goal is fairly broad, I'm not sure how to continue. I suppose I could start with a more rigorous book on real analysis, Rudin say, but I'm not sure how much overlap there is between that and Abbot. Would a book on topology be a reasonable way to go? Perhaps something else entirely? And in general, is there a reasonable progression of books to reach my goals?

I would really like to be efficient about this, since I don't have a lot of free time to spend on material besides my formal education. But, on the other hand, I do want a thorough understanding of the subject.

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