Counterexample about non Hausdorff topological vector spaces

Question

I have some troubles with Hausdorffness in TVS:

Question 1. Is there any topological vector space $X$ which is not Hausdorff?

Question 2. Give an explicit example of a topological vector space $X$ (Hausdorff / non Hausdorff) such that $(X, \tau_w)$ is not Hausdorff?

Question 3. Prove that, for any topological vector space $X$, $(X^*, \tau_{w^*})$ always Hausdorff.

Thanks in advance!

Answer 1

Q1. As mentioned in the comment, the indiscrete topology does the job.

Q2. Do you know a topological vector space $X$, where the dual space is just $\{0\}$? Then $\tau_w$ on this vector space $X$ is indiscrete.

Q3. Assume a non-zero element $x^*\in X^*$, i.e. $x^*(x)=c\neq 0$ for some $x\in X$. How can you seperate $x^*$ and $0$ by weak-star-neighbourhoods?