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I wish to learn real analysis on my own using the first 7 chapters from Baby Rudin. My goal is to have a good math background so as to study from PhD-level books in economics and finance on my own (e.g, stochastic calculus in finance, optimization, etc.). As an economics student, I find this kind of learning really challenging, so I appreciate it very much if you could give me some good suggestions/tips.

I have the following: 1) Apostol's single-variable calculus book without solutions manual 2) Spivak's single-variable calculus book + solutions manual 3) Time constraints and a rusty knowledlege of calculus (single-variable/multivariable calculus/LA)

If I have roughly 1-2 years ahead, I have to work full-time, and I have to choose only one of these 2 books, then which calculus book is better for self-study and for preparing for the first 7 chapters from Baby Rudin?

Thanks much

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    $\begingroup$ Personally, I'm a fan of Spivak. But you don't have to limit yourself to one book. You could try reading both in tandem and focus on whichever one you're getting the most out of. Also, if you're going to spend a year learning a subject, it makes sense to spend $100 to get the best book. A lot of people recommend Understanding Analysis by Abbot. It might be worth checking out. $\endgroup$ – littleO Dec 11 '17 at 9:15
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    $\begingroup$ My personal favorite is Apostol Mathematical Analysis. Spivak is good but it has too much calculus stuff also which is not really the topic covered in Rudin. If you haven't studied calculus then you can go for Spivak. I haven't gone through Apostol's Calculus so can't comment on that. $\endgroup$ – Paramanand Singh Dec 11 '17 at 9:18
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    $\begingroup$ BTW in case you do happen to read Apostol or Spivak, you will find Rudin very dry and boring. $\endgroup$ – Paramanand Singh Dec 11 '17 at 9:26
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    $\begingroup$ I would advise to say something about where you studied. As an Italian undergraduate, I always felt that Spivak's book was highly overrated: the language and topics covered have a terribly "high-school" perspective in my eyes. Rudin and Apostol are more "mature" readings. $\endgroup$ – Lonidard Dec 11 '17 at 11:58
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First of all, the writing style of Baby Rudin is very concise and it's difficult to understand Rudin's Mathematical Analysis just by reading it one or two times without previous exposure to the topic. Therefore, I advise you to stay away from that book at all costs. There are better books available these days that have been written better and offer more insights and intuition about introductory Analysis such as Pugh's Mathematical Analysis.

One particular area that Rudin's Mathematical Analysis hasn't covered well is the equivalence of sequential compactness and compactness by open covers. It hasn't talked about the Lebesgue number of an open cover at all. But that's not all of it. You want to study financial mathematics, that means most importantly you need to learn measure theory which is covered in the last chapter of Baby Rudin and it is so abstract that when you read it for the first time, you will have absolutely no idea what he is talking about. Pugh's Mathematical Analysis is more verbose and it develops your intuition when it discusses measure theory.

Also, to understand financial mathematics, I assume you have to learn about stochastic integration and Ito's formula and maybe even Malliavin Calculus. This requires a fair amount of knowledge in "Real Analysis", not "Mathematical Analysis". There are topics that are not covered in a Mathematical Analysis course such as signed measures and Radon-Nikodym Theorem.

I believe Spivak's style of writing is more geometric than Apostol in general, therefore, I believe it is better if you want to cover the topics necessary for future understanding of differential manifolds.

However, I advise you to choose Apostol's Calculus over Spivak and then continue to read Apostol's Mathematical Analysis. It is less difficult to read than Baby Rudin, it is verbose and covers a lot more details than Baby Rudin, it covers Bounded Variation functions that Baby Rudin does not cover, and it discusses Riemann-Stieltjes integral in such a detail that is way beyond someone's knowledge who has studied only Baby Rudin. It also discusses functional spaces in more details than Baby Rudin.

Therefore, my suggestion is like this: 1- Apostol's Calculus, Volume 1. 2- Apostol's Mathematical Analysis, excluding the chapters related to multi-variable analysis or Pugh's Mathematical Analysis excluding the chapter related to multi-variable calculus.

Also, I advise you to review Calculus by solving like $10\%$-$20\%$ of the problems in Apostol's Calculus, volume 1. Having learned Calculus before, and exposure to its ideas and intuitions is definitely helpful, but do not spend too much of your time (like more than $2$ weeks) on Calculus because it's really not that necessary for learning Mathematical Analysis.

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    $\begingroup$ I guess that's why you are "stressed out". :) :) $\endgroup$ – Paramanand Singh Dec 11 '17 at 12:16
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    $\begingroup$ @BlackSea: It's very common. The thing is that you're blessed to have more time to learn Calculus and Mathematical Analysis than a math undergrad. Therefore, do not hurry. As I said, if you can solve only 10% to 20% of the exercises at the end of each chapter from Apostol Vol. 1, you're fine. You shouldn't expect to be able to solve all of the exercises at first. Also, do not "cheat" and see the solution right away. I suggest you to spend 2 weeks or so on Apostol's calculus the first time, then leave it and return after a month. Mathematical maturity is slow to acquire and it takes time. $\endgroup$ – stressed out Dec 14 '17 at 4:54
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    $\begingroup$ @BlackSea: Be patient. In general, understanding a 10 page article in math might take months while understanding a 10 page article in management might take up to several minutes. Progress in math tends to be slow. I suggest you to give yourself enough time to grasp the ideas. If you don't understand something right away, do not get stuck. Give yourself sometime, accept it as a fact, move on and return to it later. If you can't solve an exercise, mark it and return to it several days/weeks later. Solving nearly 50% of the probl. in Apostol's calc. and MA is enough to proceed. $\endgroup$ – stressed out Dec 14 '17 at 5:02
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    $\begingroup$ @BlackSea: And at the end, I would like to give you a piece of advice by quoting one of the smartest people that has ever lived in this world, Jon Von Neumman: "Young man, in mathematics you don't understand things. You just get used to them." And another quote from Maryam Mirzakhani, a winner of a 2016 Fields medal: "It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out." $\endgroup$ – stressed out Dec 14 '17 at 5:08
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    $\begingroup$ @ stressed-out: I've begun to feel far more motivated after reading your comment on what it feels like if one cannot solve all or most of the exercises from a math book. I'd been feeling like I'm the only one who has such difficulty and who has to struggle through this. Again, thanks so much for your time and suggestions. It's far, far more than what I expected. Much appreciated! $\endgroup$ – Black Sea Dec 14 '17 at 7:22
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Skip the calculus texts entirely. Analysis is a formalization of a lot of the concepts from calc; in Rudin's everything is built from a foundation so prerequisites are minimal (like high school pre-calc).

I would buy Pugh's book on analysis as an alternative. It is always helpful to have a secondary source if something explained in the first isn't palatable. Good luck.

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  • $\begingroup$ But note that Calculus by Spivak develops calculus in full rigor and it serves well as an introduction to analysis textbook. It's no ordinary calculus book. $\endgroup$ – littleO Dec 11 '17 at 9:10
  • $\begingroup$ Thanks to Stephen and littleO! $\endgroup$ – Black Sea Dec 13 '17 at 23:32

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