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.