>>12440955Yeah I understand this sentiment, and I can only say it gets easier with the more math you do. You start to pick up on certain things that people like to ask about and they sort of become routine, even when the specific thing you're looking at is something you've never seen before. This makes a lot of the problems much easier. I agree that Rudin's problems are intimidating though, especially to someone not completely comfortable with proofs.
>>12441009Right so if you have linear algebra under your belt then analysis in R^n should be okay. I would pick up any standard book on the subject and just do it. I don't have any good recommendations unfortunately, but in my opinion you should stay way from Folland's "Advanced Calculus." That book gave me cancer. The theory isn't all that rich in my opinion and a lot of the mastery will come from understanding the computations. There are a couple very important theorems, like the inverse and implicit function theorems, that you should understand too, esp. if you want to study differential geometry. As for differential geometry, it depends on what you're looking for. Do Carmo has an excellent book on curves and surfaces in R^3 that really only requires linear algebra and the inverse function theorem. If you want the more advanced Riemannian geometry (modern differential geometry) I really think Spivak's "A Comprehensive Introduction to Differential Geometry" Vol 1 is the best. I loved that book, and although it's more of a smooth manifolds book, Chapter 9 I think will give you a taste of Riemannian geometry. Then you can move on to the later volumes or pick up do Carmo's "Riemannian Geometry." You don't need to do all of this (do Carmo's RG is probably fine if you have the prereqs) but you should have a solid grasp of at least the language of topology before tackling Riemannian geometry.
