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Wikipedia gives Montel's theorem saying essentially $\mathcal F$ is a normal family in $\mathbb H(G)$ iff $\mathcal F$ is uniformly bounded. My book say the same thing but iff $\mathcal F$ is locally (uniformly) bounded. Wikipedia lets $G$ be arbitrary open, whereas my book says that $G$ is a domain (connected). Presumably they are both correct. Does this mean that we can show that if $\mathcal F$ is locally bounded on a domain, then it is uniformly bounded? What's going on?

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The version of Montel's theorem I can find on Wikipedia says:

The first, and simpler, version of the theorem states that a uniformly bounded family of holomorphic functions defined on an open subset of the complex numbers is normal.

I.e. just an implication, not an equivalence. This implication is true. Clicking onward to the page on normal families, we find

Montel's theorem asserts that every locally bounded family of holomorphic functions is normal.

This is also correct. In other words locally (uniform) boundedness is enough. None of the Wikipedia pages I find say anything about the converse.

(And of course, locally uniform boundedness does not imply uniform boundedness: Take for example $f_n(z) = nz^n$ on the unit disc.)

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  • $\begingroup$ Ah, that makes sense. $\endgroup$ – fhyve Nov 19 '14 at 10:21
  • $\begingroup$ I think that the second quote lacks the word "uniformly". Because if I understand it right, the family can be locally bounded (i.e., there are bounding constants for all functions for some neighbourhoods of all ponits), but not locally uniformly bounded (there is no one constant that fits all functions for a given neighbourhood). $\endgroup$ – Igor Deruga Mar 3 '15 at 19:52

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