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This statement is clearly false: take E=K=R2, and take the group generated by the translation by 1 in x-direction, which doesn't have fixed points. So at least we need K to be weakly compact. Even with that change, I still don't see how this theorem generalizes the Brouwer fixed point theorem (as stated in Fixed point theorems in infinite-dimensional spaces), which talks about general continuous maps and not just about affine maps. I would also be much more comfortable with assuming a Banach space E rather than just a normed space in this theorem. AxelBoldt 04:04, 11 June 2006 (UTC)
Also, our article on infinite-dimensional fixed point theorems states that the Ryll-Nardzewsk theorem is from 1967, which accords with the reference I just added to the article, but does not match the 1964 Bourbaki reference given in the article. AxelBoldt 04:17, 11 June 2006 (UTC)
Oh, the "closed" should of course be replaced by compact. I remember seeing another so-called Ryll-Nardzewski theorem in Fixed point theory by Andrzej Granas, James Dugundji, which was an actual generalisation. I'll try to have a look when I have time. By the way, I was wondering : how can you have an articlemarked as a stub Chinedine 11:56, 11 June 2006 (UTC)
I'm still worried about the statement of the theorem. The Mathematical Encyclopaedia [1] requires a Banach space, but does not mention that the maps need to be affine isometries (but only that they form a "non-contracting semigroup of mappings", whatever that is) nor that the set has to be convex. Are we talking about two different fixed point theorems here? AxelBoldt 04:25, 12 June 2006 (UTC)
In fact, the existence of the Haar measure for compact groups follows from an older (and easier) theorem -- because for that application it is enough to deal with norm-compact convex sets. I think this easier fixed point theorem is due to Kakutani (although it is not what is usually called the Kakutani fixed point theorem!). This is nicely explained in Rudin's book. — Preceding unsigned comment added by 196.210.218.159 ( talk) 18:31, 4 December 2011 (UTC)
![]() | This article is rated Stub-class on Wikipedia's
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This statement is clearly false: take E=K=R2, and take the group generated by the translation by 1 in x-direction, which doesn't have fixed points. So at least we need K to be weakly compact. Even with that change, I still don't see how this theorem generalizes the Brouwer fixed point theorem (as stated in Fixed point theorems in infinite-dimensional spaces), which talks about general continuous maps and not just about affine maps. I would also be much more comfortable with assuming a Banach space E rather than just a normed space in this theorem. AxelBoldt 04:04, 11 June 2006 (UTC)
Also, our article on infinite-dimensional fixed point theorems states that the Ryll-Nardzewsk theorem is from 1967, which accords with the reference I just added to the article, but does not match the 1964 Bourbaki reference given in the article. AxelBoldt 04:17, 11 June 2006 (UTC)
Oh, the "closed" should of course be replaced by compact. I remember seeing another so-called Ryll-Nardzewski theorem in Fixed point theory by Andrzej Granas, James Dugundji, which was an actual generalisation. I'll try to have a look when I have time. By the way, I was wondering : how can you have an articlemarked as a stub Chinedine 11:56, 11 June 2006 (UTC)
I'm still worried about the statement of the theorem. The Mathematical Encyclopaedia [1] requires a Banach space, but does not mention that the maps need to be affine isometries (but only that they form a "non-contracting semigroup of mappings", whatever that is) nor that the set has to be convex. Are we talking about two different fixed point theorems here? AxelBoldt 04:25, 12 June 2006 (UTC)
In fact, the existence of the Haar measure for compact groups follows from an older (and easier) theorem -- because for that application it is enough to deal with norm-compact convex sets. I think this easier fixed point theorem is due to Kakutani (although it is not what is usually called the Kakutani fixed point theorem!). This is nicely explained in Rudin's book. — Preceding unsigned comment added by 196.210.218.159 ( talk) 18:31, 4 December 2011 (UTC)