Let be a locally convex topological vector space, with a compact convex subset .
Let be a family of continuous mappings of to itself which commute and are affine, meaning that for all in and in . Then the mappings in share a fixed point.[1]
Since is compact, there is a convergent subnet in :
To prove that is a fixed point, it suffices to show that for every in the dual of . (The dual separates points by the Hahn-Banach theorem; this is where the assumption of local convexity is used.)
Since is compact, is bounded on by a positive constant . On the other hand
Taking and passing to the limit as goes to infinity, it follows that
Hence
Proof of theorem
The set of fixed points of a single affine mapping is a non-empty compact convex set by the result for a single mapping. The other mappings in the family commute with so leave invariant. Applying the result for a single mapping successively, it follows that any finite subset of has a non-empty fixed point set given as the intersection of the compact convex sets as ranges over the subset. From the compactness of it follows that the set