An abstract general version of the Hahn-Banach theorem is shown, in the spirit of earlier works of H. König and the same author. This result is then applied to many natural situations from convex analysis, such as Moreau-Fenchel duality between convex functions and Rockafellar's surjectivity theorem for maximal monotone multifunctions in reflexive Banach spaces. It is also shown that a maximal monotone multifunction on a normed space with bounded range has full domain. An interesting feature of this approach is that it allows to bypass on several occasions the use of Brouwer's fixed point theorem.