Gorgeous. IIRC S. Eilenberg once said "Elegance in mathematics is directly proportional to what you can see in it and inversely proportional to the effort requ

Gorgeous. IIRC S. Eilenberg once said "Elegance in mathematics is directly proportional to what you can see in it and inversely proportional to the effort required to see it.". This Knill book is elegant. Topic by topic for a wide range of pure and well polished applied math, Knill is able and does go right to the most important results and for each gives a short outline of the needed definitions and of a proof, usually with good references to full details, and then states the theorem precisely. Net, for nearly any topic in pure/applied math, go there first. For many of the topics, I studied them carefully from some of the best sources, but just on a short scan this Knill book often has a better treatment, e.g., has a super nice, simple, practical statement of a fundamental theorem in Fourier series, a super nice statement of the Lebesgue decomposition from the Radon-Nikodym theorem, some nice stuff from Zorn's lemma, and much more, e.g., often some historical notes. I have to like the connection with Zorn's lemma: At one point I went to a lecture at Indiana University and to the tea before, and an older professor introduced himself as Max Zorn. Since the previous summer I'd taken a course in axiomatic set theory on an NSF summer program at Vanderbilt, I was nearly floored! All I could do was blurt out, "What did Paul Cohen prove?". The next day Zorn gave me his copy of Cohen's paper. I still have it! At one point I saved FedEx by pleasing two guys from crucial investor General Dynamics by doing a revenue projection with the differential equation y'(t) = k y(t) (b - y(t)) with y(0), k, and b given. So, the solution is a lazy S curve that starts at t = 0 at y(0) and as t increases rises, has an inflection point, and rises from below asymptotically to b. So, it is a case of growth to a market size of b. With y(t) the current revenue and (b - y(t)) the market revenue yet to be obtained, has the growth rate y'(t) proportional to the current revenue y(t) (number of current customers talking) and the remaining customers (hearing the talking) (b - y(t)). So, it's a simple model of viral growth. Well, Knill has this differential equation with b = 2 for a case of biological growth!