I am trying to show the following proposition:
if $\{ f \}$ is a locally uniformly bounded family of holomorphic functions on some domain, $D$. Then \{ f' \} is also locally uniformly bounded.
I'm assuming that a locally uniformly bounded family of functions is actually a family of locally uniformly functions, and this so-called family is actually just a set - or is it not? If so, why not call it set?