Talk:Fixed-point theorems in infinite-dimensional spaces

Linas, you wrote:

Another variant of this theorem states that if U is an open subset of C containing the origin (zero), then any bounded, contractive map f on the closure of U has one, or both of the following properties: (1) f has a unique fixed point, or (2) there is a point x on the boundary of U such that f(x) = a x for some 0 < a < 1.

I don't understand why you need U be a subset of C. It looks to me that you can get away just by requiring U to be open and bounded and not mentioning any C at all. This assuming that your C is the one from the previous paragraph.

So could you please enlighten me. Thanks. Oleg Alexandrov | talk 04:54, 3 Feb 2005 (UTC)

Because the statement applies to the closure of U ... actually, I'm not sure, I scratched my head about this as well. My first impulse is to agree with what you suggest, but I remember puzzling over this as well. The book that I was reading at the time seemed to be careful to make a distinction and also, irritatingly didn't explaining why... I kept that distinction, under the "better safe than sorry" theorem, since I already get into a lot of trouble by going too fast and assuming things that are false... linas 01:16, 8 Feb 2005 (UTC)

Do you happen to still have the book? Note that I removed that text for the time being, since I was not sure of its accuracy and since it can be easily put back if proved correct. Back then I also did not know if you got it from a book or from memory. The reason I see C as unnecessary, is because it can be always manufactured as the convex envelope of U.

If you find your book, I have another question. Is the Schauder fixed point theorem mentioned here the same as the Leray-Shauder theorem (I think I encountered that one a while ago). Thanks. Oleg Alexandrov 01:33, 8 Feb 2005 (UTC)

They're library books, and I have to make a special trip to get to the library :( That, and all the good books are always permanently checked out by someone else anyway ... I'll try to look again. linas 22:06, 8 Feb 2005 (UTC)

I will put the paragraph back again the way it is. But one day, when you run into the book, probably that is something to check. Oleg Alexandrov 23:56, 8 Feb 2005 (UTC)

Ryll-Nardzewski subsumes Kakutani?

How or why does Ryll-Nardzewski subsume Kakutani? The former is about fixed points of maps, the latter about fixed points of correspondences. AxelBoldt 05:37, 10 June 2006 (UTC)

Actually, even in the case of maps, Ryll-Nardzewski does not, as is, subsume Kakutani; indeed, Kakutani is about all compact convex sets in a locally convex (Hausdorff) TVS, whilst Ryll-Nardzewski is only for the weakly compact in a normed space under isometries. The difference is huge (and accounts for the whole notion of amenable groups). I'll delete the wrong statement.128.178.14.162 (talk) 07:55, 30 June 2008 (UTC)

The statement of Schauder's Fixed Point Theorem

In the statement, it is said that the image should be countably compact. But we are in a Banach space here, which is a metric space. In a metric space, the notion of countable compactness is equivalent to the notion of compactness. Right? So it's enough to say that the image should be compact. /Janzon

Yes, I thought that was strange too - nobody talks about "countably compact" sets in a metric space (except to point out that they are just the compact sets). So I've changed it. --Zundark 17:22, 23 November 2006 (UTC)