# Does there exists a non-Hausdorff topological vector space with a proper convex open subset?

Wikipedia's page on the Hahn-Banach theorem mentions a technique that converts a non-Hausdorff topological vector space $$X$$ into a (Hausdorff) locally convex one: apply the weak topology (induced by $$X^*$$). However, the argument assumes that $$X$$ contains a proper, convex, open set.

The standard example of a non-Hausdorff topological vector space is $$\mathrm{Spec}{(\mathbb{R}[\vec{x}])}$$ with the Zariski topology. This does not contain a proper open convex subset.

What is an example of a non-Hausdorff topological vector space to which this technique applies?

• The Zariski topology does not make $\mathbb{R}^n$ a topological vector space. Addition is not continuous. – Eric Wofsey Oct 6 '20 at 3:57
• In fact, every non-Hausdorff topological vector space has the form of the example in your answer: it is a product of a Hausdorff topological vector space with an indiscrete vector space. – Eric Wofsey Oct 6 '20 at 4:00
• @EricWofsey: (#1) Oops. I should have known better --- this sort of thing is why we have schemes. My algebraic geometry is getting rusty, I guess. Thanks. – Jacob Manaker Oct 7 '20 at 5:33
• @EricWofsey: (#2) That sounds like it would make an interesting answer.... – Jacob Manaker Oct 7 '20 at 5:34

Yes, there is! Let $$\mathcal{T}$$ be the indiscrete topology on $$\mathbb{R}$$ and consider $$X=\mathbb{R}\times(\mathbb{R},\mathcal{T})$$. This is a topological vector space (exercise); and $$(-1,1)\times\mathbb{R}$$ is an example of a proper convex open subset.

1. $$X$$ has a topology induced by the seminorm $$\|(x,y)\|=|x|$$.
2. $$X$$ is locally convex (even without the "weak topology trick").
3. $$X^*\cong\mathbb{R}^*\oplus0$$; all continuous functionals on $$X$$ are constant on the second direct factor.

I leave these claims, as well as the construction of an infinite-dimensional example, to the interested reader.

Finally, a general remark: the Zariski topology is much more pathological than one needs to obtain non-Hausdorffness. You should maybe consider re-reading Stein and Seebach.