Talk:Equicontinuity
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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)
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- Andrew, thanks for your changes. I would be interested to know whether the theorems in the article are also valid in this general setting.
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- 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.
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- 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.
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- Jitse Niesen 11:50, 7 Apr 2004 (UTC)
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- I now replaced "equally convergent" per Andrew's suggestion. -- Jitse Niesen 17:44, 10 Apr 2004 (UTC)
