TITLE: An counterexample of Hahn-Banach theorem in a topological vector space
QUESTION [5 upvotes]: Problem : Give an example of a TVS $\mathcal{X}$ that is not locally convex and a subspace $\mathcal{Y}$ of $\mathcal{X}$ such that there is a continuous linear functional $f$ on $\mathcal{Y}$ with no continuous extension to $\mathcal{X}$ 
I think this problem means that Hahn-Banach theorem ( LCS version ) may not hold in a TVS.
But I can't find a counterexample..

REPLY [3 votes]: Hint: Do you know of a TVS with trivial dual space? Then take any non-zero linear functional on a one-dimensional subspace.