Talk:Normal family

Clarification on the assumed distance metric

I added a clarification about the assertions relating to complex analysis and Cauchy's theorem. The clarification is that the assumed distance metric is $d_{Y}(y_{1},y_{2})=|y_{1}-y_{2}|$.

I also observe that the statement of Montel's theorem here is different from the one in the Montel theorem article. That statement assumes the functions are defined over an open set, and requires uniform boundedness. I wonder if some more detail is needed here.

Continuity

Since the convergence is uniform, need we say "to a continuous function from X to Y?" If the function from X to Y were jump discontinuous, convergence could be at best uniform a.e. right? Could you have jump discontinuous functions in the family?

168.105.250.112 (talk) 08:46, 5 January 2012 (UTC)

Montel discussion long ago

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)

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)

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)

Naming

I disagree with the content of this paragraph. In "modern terms", one can just say that it is a relatively compact set for the compact-open topology. — Preceding unsigned comment added by 2A01:E34:EE33:210:596:72BD:DC4C:C9FA (talk) 18:43, 13 February 2014 (UTC)

Definition when Y is a metric space in the lead

This definition seems off to me. It seems to be describing a condition for compactness, rather than pre-compactness. Functions should be allowed to "diverge locally uniformly" (def. from p. 33 of Milnor) Student298 (talk) 00:05, 5 March 2020 (UTC)