The ambiguity between 
as a vector space in its own right and as mere shufflings about of scalar random
variables is very difficult for me to settle. It was clear early on that the latter approach would be the easiest
to treat rigorously. I really wanted to attempt the former, and think of everything as vectors and matrices. Of course,
one builds the former out of the latter, but one can't also know everything when one begins. I have kept
things as they stand as a challenge to myself, because I know the development can proceed this way, and because
I think it makes the topic more approachable to non-specialists.