Real Analysis : Self Studying vs Doing a Course

I am an engineering graduate student. Recently I got interested in studying Maths. So, I have started self-studying Real Analysis(let's call it RA) using a few books. I will also be using problem books to supplement my learning.

I told my adviser that I'd be devoting a certain amount of time in studying Maths, apart from the time that I devote to my engineering thesis.

He told me that self-studying RA would not be a good option. RA is something which requires the discipline of a coursework. You might get some feel of it but the rigorousness can only be achieved through a course. He said that pure mathematics is not something like programming in which you can become master by self-studying.

I did not debate him on this. But I wanted to ask you this:

If I am extremely dedicated to studying RA, devoting 4-5 hours daily and doing problems given in the textbook or a separate problem book, trying to understand every theorem/concept which the author says is important, and honestly attempting problems, will I be able to understand Real Analysis in the same manner as doing a course. My target is to learn RA up to the level of an Undergraduate Mathematics Major.

Your answer should answer some or all of the following questions:

• Is it possible at all to achieve the same level of understanding through self-studying as compared to a course?
• What is the essential difference (the "gist") between self-studying and doing a coursework in RA?
• How much more time should it take in self-studying, comparatively?
• Can honestly attempting problems in the books be an alternative to exams and assignments?

PS: I'm in my masters right now but will be going for PhD which will also be in Engineering. However, during PhD also I'm thinking of honing my Maths skills and might go for a separate major later on if I so desire. So, essentially I am hell-bent in understanding Mathematics which has intimidated me for so long. And as it is turning out, I am actually enjoying it thoroughly!! So that's that.

Solution 1:

I think it is certainly possible. But you should have someone check your exercises. Some books do not give solutions or sometimes it might be not very clear for a beginner in analysis if his solution is the same as the books. Analysis is usually the first lecture where one lerns to do mathematical proofs and it should be lerned and done in the correct way. You should at the end be able to self check yourself and be critical enough of your own work to see if it is correct or not. Also you should lern to write down your proofs and ideas in a formally correct and comprehensible way. I suggest the best way to learn this is write your lecture notes and exercise solutions down. Preferably using latex and re read and rethink them again over the course of your study. Also if possible give them to someone to check.