I’ve found the concept of dual spaces in Linear Algebra just the dizzying height of abstraction. Already the very object of study, the “vector space”, is an abstraction that most people find confusing on first meeting it. It’s been years since I first encountered such an abstraction so by now I don’t bat an eye at it. However, we then discuss linear functions between vector spaces … ok, I’m pretty good with the idea, I’ve had a lot of practice with this sort of stuff. Then we talk about linear functionals on vector spaces, a particular kind of linear function. Again, I’m good with that. Then we talk about the dual space, the space of all functionals … whoa, starting to get wobbly. A vector space that you construct out of functions of vector spaces?
… Alright, that was kind of a blow, I think I can handle it. But let’s start moving slowly because this is weird territory. Next we get the dual transformation of any given linear function, and now it feels to me that all hell has broken lose. A transformation of a space of transformations of a space to a space? Oh dear god what are we doing?
So that bothered me about six months ago. Since then I’ve gotten distracted by certain programming topics, and now that I’ve figured out a lot of what I wanted from that subject, I’ve been returning to the Linear Algebra. I dreaded coming back to the dual transformation. But as I went carefully through it again, taking careful notes and trying to solve problems before reading the solutions in the textbook, I found a remarkable new level of comfort with everything. Sleeping on the subject helped a great deal.
In fact it’s a somewhat surprising common-place how a subject that confounds you one day, the next morning can seem simple. I can’t tell you then number of times I’ve woken up with a solution to last night’s problem. Sleeping on a problem is a tried and true technique of proof for me, at this point. Like I think Von Neumann said, “In Mathematics there is no understanding, there is only getting used to.”