Linear bijection non-preserving Hausdorff propery

Question

My question is: If $f: X \to Y$ is a continuous and linear bijection between topological vector spaces, is it possible that $X$ is Hausdorff and $Y$ is non-Hausdorff? (TVSs are considered in the more general sense here -- they need not be Hausdorff.)

Answer 1

Let $(X,\tau )$ be any Hausdorff topological vector space , and let $\sigma =\{\emptyset , X\}$ be a trivial topology on $X.$ Then the identity $\mbox{id}_X :(X,\tau )\to (X ,\sigma )$ is a continuous and linear bijection but $(X,\sigma )$ is not Hausdorff topological vector space.