Is there an example of a, non-Hausdorff, topological vector space which has bounded subspaces

Question

Let $E$ be a topological vector space which is not Hausdorff.

Is it true that every non-trivial subspace $M$ of $E$ is necessarily unbounded?

Or, does there exist, non-Hausdorff, topological vector spaces containing non-trivial bounded subspaces? If any, what would be a counterexample?

Thanks in advance.

Answer 1

Take $\mathbb{R}^2$ with the semi-norm $$\|(x,y) \| := |x|$$ and the subspace $$Y:= \{(0,y) : y \in \mathbb{R} \}, $$ then $( \mathbb{R}^2, \| \cdot \|)$ is a non-Hausdorff TVS with bounded subspace $Y$.