This is my current situation: I have just finished my real analysis lectures (measure theory and differential calculus in several variables) and I am looking forward to take functional analysis next. I have time now to do a bit of self-study to prepare for this course. I am particularly interested on a generalization of the notion of differentiability taught in real analysis courses. So I did some research. I heard of
- Gateaux-differentiability
- Frechet-differentiability
Since I want to do it right from the beginning, here is my question.
What is a good and usefull generalization of the notion of differentiability on $\mathbb{R}^n$ in functional analysis (or maybe there is some other field)? Does someone have a good book for studying this?