Talk:Equicontinuity

Suggestions for improving the article:

• Add some examples of families that are or are not equicontinuous.
• Having written the examples, we could try to merge them in an informal introduction without epsilons and deltas.
• How far does Ascoli's theorem generalize? Is it valid for all functions on compact metric spaces?

Jitse Niesen 16:22, 8 Feb 2004 (UTC)

There's a very general version (Dieudonne Foundations of Modern Analysis 7.5) for functions on a compact metric space, values in a Banach space.

Charles Matthews 12:00, 9 Feb 2004 (UTC)

I see that the page is really about equicontinuous sequences. I had always taken equicontinuous sets of functions to be the prime notion; as in characterising relatively compact subsets in function spaces.

Charles Matthews 08:28, 7 Apr 2004 (UTC)

I agree - I added the most general version (topological spaces, uniform spaces) at the end the other day. I think a mod of the opening par (drop "sequences", put in context - compactness, Banach-Steinhaus theorem, etc) would improve things. My second-last definition should also be modified to include "the set of functions A is equicontinuous" = "for all x, the set of functions A is continuous at x". Be my guest. -- Andrew Kepert 08:40, 7 Apr 2004 (UTC)

Actually, the opening par is abysmal -- wtf does "equally convergent" mean? I would instead say "have equal variation over a given neighbourhood", or something similar. -- Andrew Kepert 08:45, 7 Apr 2004 (UTC)

Andrew, thanks for your changes. I would be interested to know whether the theorems in the article are also valid in this general setting.

I wrote the opening paragraph with the idea that it should be understood by as many readers as possible. I agree that "equally convergent" is a horrible phrase, but I couldn't think of anything better.

The concept of equicontinuity can be used on various levels of sophistication. The article as I wrote it, is the simplest level: equicontinuity in real analysis, as taught to first-year or second-year undergrads. For this, it seems that you only need equicontinuous sequences, because you basically want to establish a connection with convergence. Of course, equicontinuity can also be used in the more general context of functional analysis, and in this setting, we should talk about the connection with Banach-Steinhaus, compactness, etc. I did not do this because I am not confident enough of my knowledge in this area. This would certainly be a valuable addition to the article, and I would be grateful if somebody would write along these lines; however, I do feel that part of the article should be in a low-brow style.

Jitse Niesen 11:50, 7 Apr 2004 (UTC)

I now replaced "equally convergent" per Andrew's suggestion. -- Jitse Niesen 17:44, 10 Apr 2004 (UTC)

Please add another definition of equicontinuous. I never really got confortable with the epsilon-delta definition of continuous. --151.204.141.69 13:05, 20 April 2007 (UTC)

There's a comment (along with some references, but they are not online references) that a Fatou set is equicontinuous and a Julia set is not, but this comment makes no sense as stated, since the Fatou and Julia sets of a function are sets of points rather than sets of functions. I could hazard a guess that it intends to say that the set of iterates of a function are equicontinuous at points in its Fatou set, and not at points on its Julia set... which would at least type-check, but I can't be sure that my guess is actually right. If someone has access to the listed references, it would be good to change it so that it at least makes sense. Cgibbard (talk) 08:51, 31 August 2009 (UTC)