It should be clear that if is holomorphic then for any we have is holomorphic, so if is bounded, and since is bounded iff it sends bounded sets to bounded sets, then is bounded so Liouville's theorem applies and gives constant for all . Assume is not constant ie. there exist such that then there is a functional that assigns to a non-zero complex number (Hahn-Banach), so if are linearly dependent we're finished, if not apply Hahn-Banach again assaigning any other non-zero complex number. Either way we contradict the fact that is constant.