Topological Vector Space is separable if its dual space $X^*$ is separable?

Question

Let $(X,\tau)$ be a Topological Vector Space such that the associated dual space $X^*$ is separable. Can we say that $X$ is separable ?

I know that this property is valid for Banach spaces but for topological vector spaces, I have no idea.

An idea please.

Answer 1

This can fail badly if $X$ is not locally convex. Consider $X = L^p(\Omega,\mu)$ where $0 < p < 1$ and $\Omega = [0,1]^\Gamma$ where $\Gamma$ is uncountable and $\mu$ is the product Lebesgue measure. Then $X$ is non-separable, but $X^* = 0$.