I am studying the Hahn Banach Theorem in the case of Topological Vector Spaces, and would like to prove the equivalence between the following three statements:
[(HB1): Geometric Hahn Banach Theorem] Let E be a TVS, M a linear manifold in E and A a nonempty convex open subset of E such that $M\cap A=\emptyset$. Then there exists an hyperplane containing M and not intersecting A.
[(HB2): Analytical Hahn Banach Theorem] Let E be a vector space, p a seminorm on E and M a subspace of E. If f is a linear form on M such that $|f(x)|\leq p(x)\ \forall x\in M$ then there exist a linear extension to E such that $|f_1(x)|\leq p(x)\ \forall x\in E$
[(HB3): Separation of Points] If E is a Hausdorff LCS, then E' separates points of X. That is, if x and y are distinct points of E, then there exists some $\lambda\in E' $ such that $\lambda(x)\neq\lambda(y)$.
Above E' is the continuous dual of the space E.
I have understood the proof of (HB1) based on the Zorn's Lemma.
Moreover in many references (Treves, Schefer) one can find a proof of $HB1\rightarrow HB2$. Schecter which claims the three result equivalents, does not provide explicit proofs of this implications, since he actually goes through various other equivalent results.
Does anyone know a reference or may suggest how to see the remaining implications?