I'm an undergrad currently taking my first course in QM. We've just about finished our chapters on the mathematical formalism and it has been making me pose a lot of questions. We use a lot of vocabulary/definitions that I suspect come from higher algebra, topology, & functional analysis all which are alien to me at this stage in my curriculum. Terms like compactness, completeness, orthogonal functions to name a few, I feel like am expected to take them on faith without really knowing what they mean/why they're used in QM (I can see the parallelism between the cartesian unit vectors i,j,k and cos/sin functions in the interval [0, 2pi], but that's about it). Our suggested textbook (Cohen's) doesn't aid me much other than giving me more precise definitions of what is talked about in class. I pretty much felt the same way in analytical mechanics when we covered Hamilton-Jacobi theory and my last MM course on integral transforms. Integral transforms was taught in a plug n chug manner and I didn't quite know what significance swapping a problem into a "reciprocal space" had, other than to make some PDE/ODE problem simpler... Am I getting ahead of myself? Is there anything I can do that can remedy my situation? I get the impression that this gets even worse in more advanced subjects like QFT, which I would really like to take at the grad level. Should I just worry about completing the course in QM before pondering all of this? Feel free to shake the curiosity out of me if you feel I risk failing my course from spending time on these questions haha.