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I'm a first year graduate student in physics who picked up both an undergraduate degree in math and physics. However, in getting the math degree there were a few courses I either didn't take or simply didn't feel I had learned to the fullest extent. So, to remedy this I'm looking for some book recommendations on the following subjects:

1: Functional Analysis. Never took this course and am looking for two types of books: a mathematical type text that may be done at say the third or fourth year undergraduate level, and a more physics applications oriented one. For example covering Sobolev spaces.

2: Complex Analysis. I only want to look into a more mathematical text for this one, and was thinking Ahlfors but am open to suggestion.

3: Algebraic Topology. Self studied mostly from a collection of different sources and then Hatcher. Was looking for a physics oriented text to go along with this.

Thank you.

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  • $\begingroup$ I don't have strong preferences for 1. or 3... certainly Hatcher is fairly popular and not a bad choice. I just taught complex analysis last year and the top of my text list goes like: Saff and Snider. Marsden, Gamelin, Alhfors, Albowitz and Fokas. The text by Saff and Snider is the most computational clear. It reads like a low-level calculus text in places. Gamelin has excellent insights and some deep work on green's functions, and Albowitz and Fokas has a lot of breadth. Alhfors influences all the others. I can't choose. I like them all. $\endgroup$
    – James S. Cook
    Sep 8, 2013 at 3:24
  • $\begingroup$ For functional analysis, mathematical and at the 3rd to 4th year undergraduate, look at Introductory Functional Analysis with Applications by Erwin Kreyszig. $\endgroup$
    – Dave L. Renfro
    Sep 11, 2013 at 19:23

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1: I would recommend 'Linear Functional Analysis' by Rynne and Youngson. I'm not sure about the book by Kreyszig, as it seems like it goes off on a tangent quite a lot and gets a bit crazy in places. For functional analysis with physics applications, see Barry Simon.

2: For complex analysis I'm not sure, I did study it but we had our own set of coursebooks from the university which were basically the only thing we used. Maybe the book by Ian Stewart, but I haven't read it.

3: As you have a background in physics, I recommend 'Introduction to Topological Manifolds' by John Lee before Hatcher. Maybe try Munkres if you find Hatcher not to your liking, as I believe the content is similar. For physics, the book by Nash and Sen is well-known, although I must admit I haven't actually read it.

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