Thanks CompuChip. I'm still confused and most of what you told me is stuff that I already know, however some of your terms were helpful. The notion of a 'metric space' and 'inner product space' for instance. I see now that I had the concepts of coordinate system and metric nearly equated...
I'm trying to build up enough understanding to work through some GR on my own, but I'm horribly confused by some of the math concepts. So terribly so, that I'm not even sure how to ask my questions. Please bear with me.
Lets work in a 2D plane.
Assume I have a vector u which I can write out...