Some More Applications of the Hahn-Banach Theorem

We give a new proof of the fact that equivalent norms on subspaces can be extended. This new proof is based on the Hahn-Banach Extension Theorem. We also give new characterizations for an equivalent norm on a dual space to be a dual norm. Finally, a new proof of a particular case of the Hahn-Banach Separation Theorem is provided without involving the Axiom of Choice.