I have a question for the mathematical community about the language mathematicians use for a series of functions, and it has been bugging me for a while. I was thinking of posting it on MathOverflow, but it seems to fit the “subjective and argumentative” criterion for closing rather well.
Suppose someone introduces a countably infinite set of holomorphic functions on some open subset U of the complex numbers, and wants to argue that the
sequence sum converges to a holomorphic function. One valid way to prove it would be to show that the sequence sum converges locally uniformly absolutely (or uniformly absolutely on compact subsets of U), and then point toward the complex analysis text of choice.
Question: If someone only argues that the
sequence sum converges absolutely, should I complain? (If so, how loudly?)
I tend to think that absolute convergence is a pointwise statement, and doesn’t a priori tell me anything about holomorphicity. On the other hand, there is a theorem of Osgood (Ann. Math. 1901) stating that pointwise convergence of a sequence of holomorphic functions to a function implies that there is an open dense subset V of U, such that the function is holomorphic on V, and the convergence is uniform
ly absolute on compact subsets of V. While this theorem is rather impressive, it doesn’t give me quite what I want, since V is not necessarily all of U (and there are interesting counterexamples, e.g., in this expository pdf). Unfortunately, a sample of published papers that use holomorphic functions reveals that people use the absolute convergence language a lot. This leads me to consider the following two (non-exclusive) possibilities:
- The term “absolute convergence” has become an abbreviated form of “uniformly absolute convergence on compact sets” among people who know what they are doing.
- Some mathematicians are rather sloppy when discussing convergence, and this includes referees.
Any input would be greatly appreciated, especially input from those more analytically experienced than I.
[Apologies for any confusion caused by the previous version.]