An counterexample of Hahn-Banach theorem in a topological vector space

Question

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..

Answer 1

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