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Linas, you wrote:
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:::: 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. [[User:Oleg Alexandrov|Oleg Alexandrov]] 23:56, 8 Feb 2005 (UTC)
== Ryll-Nardzewski subsumes Kakutani? ==
How or why does [[Ryll-Nardzewski_fixed_point_theorem|Ryll-Nardzewski]] subsume [[Kakutani_fixed_point_theorem|Kakutani]]? The former is about fixed points of maps, the latter about fixed points of correspondences. [[User:AxelBoldt|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.[[Special:Contributions/128.178.14.162|128.178.14.162]] ([[User talk: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. --[[User:Zundark|Zundark]] 17:22, 23 November 2006 (UTC)
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