Conway's functional analysis proves the existence of Haar measures. One of the points of proof is to utilize the Ryll-Nardzewski fixed point theorem, which states that a semigroup of weakly continuous affine maps defined on the weakly compact convex subset of locally convex space has a fixed point if it satisfies the nonctracting condition.
However, in the proof, I confirmed that the weak-star continuous affine map defined in the weak-star compact set is utilized, not the weakly continuous affine map defined in the weakly compact set. Is this an error in Conway's book, or is it because the Ryll-Nardzewski theorem still holds for weak-star topology?
** The locally convex space in the proof is a space of complex regular Borel measure, where the weak topology does not coincides weak star topology, since it is the dual of continuous functions, which is not reflexive Banach space in general.