Talk:Normal family

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Is the theorem here Montel's theorem? Certainly Montel did a lot with normal families.

This material is used in complex dynamics, so perhaps that connection should be made.

Charles Matthews 13:56, 8 Feb 2005 (UTC)

First, thank you for your changes, I had forgotten to mention that normal family is a math concept.

I don't quite know these things. So I saw the page on complex dynamics but I don't know what to add to it. And neither do I know about Montel's theorem (somehow I don't remember it from the grad course in complex analysis I took a while ago).

So what you are dealing with here, is a janitor who moves stuff from planetmath without knowing what I am doing. :) Oleg Alexandrov 15:55, 8 Feb 2005 (UTC)

http://planetmath.org/encyclopedia/MontelsTheorem.html is the PM page on Montel's theorem. It is about normal families, so could be added to this page.

Now to look up Montel. Charles Matthews 16:08, 8 Feb 2005 (UTC)

This biography page is quite useful: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Montel.html. Charles Matthews 16:12, 8 Feb 2005 (UTC)

I will look at these later today. Oleg Alexandrov 17:36, 8 Feb 2005 (UTC)

[edit] Montel

One note I have, is that the last sentence you put in the first paragraph needs some editing I think. You see, the point of the paragraph is to introduce a normal family of functions. But then you switch to talking about compact sets in function spaces. I of course understand what you mean, but the change is too abrupt I think, one would need to make the connections:

Normal family -> precompact set -> compact set in a function space

which could be too much for an introductory paragraph and a newcomer to these things. I would suggest that it be developed into a gentler standalone paragraph somewhere below. But it is up to you.

Now, I copied Montel's theorem over. (And thus I read it.) I think what is mentioned in this normal family article is not that theorem. This article just says that a sequence of holomorphic functions, that converges uniformly on compact sets, converges to a holomorphic function, while Montel's theorem has the much stronger statement that a locally bounded sequence of holomorphic functions has a subsequence which converges to a holomorphic function.

One weakness of PlanetMath and which we copy over here, is that their articles have very little motivation and connections to other topics. I will think of what else to add to this and to Montel's theorem article. Suggestions (and actual edits) welcome. Oleg Alexandrov 05:08, 9 Feb 2005 (UTC)

[edit] Correct definition

Yes, there is some problem with their way of doing things. I wasn't aware that 'normal family' was a concept used outside complex analysis. Also, equicontinuity is a related but different concept. So it would be good to think how to integrate this material, better. Charles Matthews 07:44, 9 Feb 2005 (UTC)

I think you are right saying that the definition is not used outside complex analysis. The reason they make things more general, is, they way I see it, because they also talk about holomorphic functions defined on the Riemann sphere in addition to the complex plane.

That said, I incline to agree with you that they generalize excessively. Should I remove the general definition, and the case of functions on the Riemann sphere, and just keep the case of holomorphic functions in the complex plane? Oleg Alexandrov 03:21, 10 Feb 2005 (UTC)