The Hahn-Banach theorem: a proof of the equivalence between the analytic and geometric versions (Q2657343)

The main goal of this short note is to present a simple and original proof of the classic geometric version of the Hahn-Banach theorem in the general context of locally convex spaces, deducing it from the analytic version. As is well known, the complex case is a consequence of the real one, thus the author deals only with the latter. The use of a seminorm seems to be essential, since examples are provided illustrating that the seminorm obtained in the proof cannot be a norm.