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)[reply]