# 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?

• Hi Tom! Do you have suggestions of what kinds of things a HB separation theorem on Hilbert space would accomplish that can't be done in Banach space? Feb 13, 2014 at 16:54
• Hi @ABlumenthal! Orthogonality of the separating hyperplanes to the distance-minimizing vector, relationship between the metric distance and the marginal difference between the hyperplanes, etc. Feb 13, 2014 at 17:23
• Riesz representation. Feb 13, 2014 at 18:05
• Sorry, @Michael, that is not helpful. Could you expand on what you mean? The Riesz representation theorem is not a form of the Hahn-Banach separation theorem, though it is useful in stating it. Feb 13, 2014 at 19:31
• I guess the proof of the usual separation theorem would be easier because you don't have to appeal to Hahn-Banach. 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 $a$. I think the linear functional $\ell(y)=(y,x-a)$ separates $x$ from $A$. Sep 27, 2018 at 17:38

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.