Talk:Normal family
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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 .
I also observe that the statement of Montel's theorem here is different from the one in the
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)