How important is the choice of books in studying Analysis?

Question

I am in a fix. I have done a graduate course in Pure Mathematics.I love to study abstract algebra.I want to do postgraduate in Mathematics especially in Abstract Algebra .

In order to enter a postgraduate program I have to qualify a screening test which comprises of three sections:

1. Algebra
2. Analysis
3. Metric Spaces and Topology

I appeared in the test the earlier year .Though I scored a perfect 10 in Algebra,my marks in the other two topics were $3$ and $4$ out of $10$ as a result of which I failed to qualify.

Is it possible to learn Analysis now or its too late.Can I speed up the process of learning Analysis?Can someone please give some tips on how should I read this topic?I don't know why I fail in this topic very badly.Though I have studied topics like Continuity,Differentiablity,Riemann-Integral,Sequence and series of functions etc.,I fail in these topics miserably.Do I need to start from scratch now?

Also I more question I find that people ask for recommendation of books in Analysis.How important is the choice of books in a particular topic.We followed Rudin-Principles of Mathematical Analysis in our course.

In short please give some tips on how to study Analysis .I know there are many on this site who are very good at analysis.Any recommendations will be helpful

Answer 1

Since you are talking about the fundamentals, for the one-variable case I can only recommend Spivak's excellent book "Calculus". I learnt a lot from it even before I entered university and it has provided me with intuition which has been very helpful ever since.

For multivariable calculus things aren't so clear for me. I'm not really sure whether there exists a vector calculus textbook which is actually good, since I learnt from lecture notes and solidified my knowledge from experience in electromagnetism and differential geometry (btw if you're into that, you may benefit from O'Neill's "Elementary differential geometry"). At any rate, I enjoyed Marsden & Tromba's book "Vector Calculus", which is however not very rigorous. You can again go for Spivak's little monster called "Calculus on Manifolds", but be warned: the notation is kind of irritating from time to time and it is more oriented towards geometry, not to mention that it is extremely laconic compared to his other books.

Rudin's book is great but in my opinion it can only be of use if you are exceptionally good or have already a solid background in analysis. A fantastic book which has a lot of overlap with Rudin is Apostol's "Mathematical Analysis", which covers even more material but in a much friendlier manner. I guess this last one is probably what you are looking for.

Finally, concerning metric spaces, I'd suggest Kaplansky's "Set Theory and Metric Spaces", the "metric spaces"-part of which has appealed to me ever since I read it for the first time. He has very well placed definitions and really enlightening exercises. I'd also consider Simmons' "Introduction to topology and modern analysis", for which I have a soft spot as it has been the first book I read metric space theory from. His language initially seemed kind of strange but I finally realized how fitting and natural it is.

I think that I would begin with Spivak, then move to Marsden&Tromba for a little while and then to Apostol and Kaplansky.

Now, concerning "strategic" advice, I can't say much since you are not providing sufficient information on your background, but for some reason my guess is that your exams were less computational and more "theoretical" than you expected. In this case, I'd say you can take Apostol's book for example and fill every proof you can on your own. If you do this I believe that you'll be more than ready (:

Good luck with your tests!