I read the proof of the Hahn-Banach extention theorem on wikipedia and see no mention of the axiom of choice. The proof goes by constructing an appropriate convex cone and then invoking the Riesz extension theorem, whose proof is based on transfinite induction. However, I've heard several times that proving the Hahn-Banach extension theorem (for topological vector spaces) requires AC or some slightly weaker version (ultrafitrations). I conclude that if the proofs on wikipedia are correct, then there must be a disguised use of AC somewhere (in Riesz of Hahn-Banach extension theorem).
Question
So, where is AC used in the proof of the Riesz or Hahn-Banach extension theorems ?