Sorry, I can't recommend anything. The course material and even the textbook I read on the subject was in German. Yes, constructs are not necessarily applicable, but it is quite hard to find any mathematical construct that has not yet been used (or abused) for some application outside of pure mathematics. This careful reasoning about convergence is exactly what physicists usually don't bother with. Until proven otherwise we just assume it converges and go right ahead. If we notice along the way that it doesn't, we try another approach, but as long as a technique works we don't care much about proving that it is applicable.