Talk:Ryll-Nardzewski fixed-point theorem


 * if E is a normed vector space and K is nonempty convex subset of E, which is closed for the weak topology, then every group of affine isometries of K has at least one fixed point.

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 stubChinedine 11:56, 11 June 2006 (UTC)


 * Stub'd. (Added topology-stub, but mathanalysis-stub may be more appropriate.)  &mdash; Arthur Rubin |  (talk) 16:17, 11 June 2006 (UTC)

I'm still worried about the statement of the theorem. The Mathematical Encyclopaedia 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)
 * See Math project talk page where I saw and replied to your query.--CSTAR 12:42, 12 June 2006 (UTC)


 * Ok, I copied your reply here, to keep it all in one place:
 * As stated it's wrong. The semigroup is required to satisfy another property, that it be "distal". Also I don't think it can be used to prove existence of Haar measure on general locally compact groups, although I think for compact groups yes.  I think this is in Rudin's functional analysis book for instance.  Also see Frederic Greanleaf's little book (now horribly outdated) on "Amenable Groups".--CSTAR 12:41, 12 June 2006 (UTC)
 * I take this back. It's correct. I noticed that it says "affine isometries" which I had missed. A semigroup of isometries on a metric space is automatically distal. The assumption requiring the normed space E to be Banach is not necessary as the following argument shows:  Complete the underlying space E; the dual space of the completion is the same as that of E, and the inclusion mapping is continuous with both spaces E and its completion equipped with the weak topology. Therefore the original compact convex set is still compact (and convex of course) in the completion.


 * However the claim regarding Haar measure is true only for compact groups to the best of my knowledge.--CSTAR 03:04, 8 July 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)