This has been the most involved intellectual project of my life thus far, and has been frought with supreme enjoyment and abject frustration - generally both at the same time. While the main body of the paper is certainly not free of my personality, I think it is important to make explicit here at the end some of the issues that have been raised by this. In other words, I want to locate myself in relation to the paper, and to my further goals.
I began this paper with no knowledge of Measure Theory (and hence, of rigorous probobility theory), Hilbert Space Theory (at least of the infinite dimensional variety), and no real understanding of the importance and nature of orthogonality and projection. While this ignorance is sure to be obvious to any more knowledgable reader, I do feel that my knowledge of these subjects has grown considerably in the course of writing this beast.
I would also like to express my feeling about the strengths and weaknesses of the paper as I see them. I think the projections/Hilbert Space approach is a good one, in that it yields a simple, clean derivation which is comprehensible intuitively in familiar geometric terms. While I can hardly claim that my work here is truly original, I do feel a great deal of satisfaction in knowing that no book I found presented exactly this approach. Most make some offhand comments about geometric interpretations, but end up deriving it in some ugly fashion which hides the essential simplicity and beauty of the theory. I am very proud of the fact that I knew how I wanted to understand it, and despite huge gaps and mistakes, I feel that I have achieved this.
My main dissapointment is in not having been able to incorporate Information Theory.
This was my original intention, but in the end it was enough to just sketch what I have here. Between comments in Kalman's paper, and my own research, it is clear that there are deep, important connections to be made here. Fisher Information, the Cramer-Rao lower bound, the Gaussian Channel, and Mutual Information all have a great deal of relevance here, and I dearly hope to be continue my work in this direction.
In the end, as the culmination of a Bachelor's Degree in Mathematics and a big step down a beautiful path, all I can say is Thank You to all the people who have helped me achieve this. I believe I've actually done it. I cannot believe how much I've learned, and how much more I've learned there is to learn.