Hahn-Banach separation theorem for Hilbert spaces What is the strongest form of the Hahn-Banach separation theorem for Hilbert spaces? Could you please provide a reference?
 A: This doesn't answer the question of giving the strongest Hahn-Banach separation theorem for Hilbert spaces. But it does address a closely related question of how to easily get separation in Hilbert spaces, without needing Hahn-Banach. So maybe the method suggests how to get a really stronger separation theorem in Hilbert spaces.
If $A$ is a nonempty closed convex set in a Hilbert space and $x \in X$, then there exists a unique closest element of $A$ to $x$, call it $a$. The linear functional $\ell(y)=(y,x-a)$ separates $x$ from $A$. I think you can use this idea to separate $A$ from other sets besides singeltons $\{x\}$.
Edit: Let me explain how I came up with this. I drew a picture in $\mathbb{R}^2$ with a convex set $A$ and a point $x$ not in $A$. I drew the shortest line from $A$ to $x$. That's the vector $x-a$. Then I drew a second line orthogonal to the first that separates $A$ from $x$. The second line is $\{y: y \cdot (x-a) = c\}$ for some $c$. In other words, the second line is $\ell(y)=c$. One reason Hilbert spaces are great is that you can do exactly same thing.
