Q1. Does there exist a separable Banach space $X$ satisfying in the following property?


1- $X^\*$ is non separable. 


2- For every countable subset $F\subset X^\*$ there exists $0\neq x\_F\in X$ such that 
$f(x\_F)=0$ for all $f\in F$. 


Q2. If it is impossible, what about if we replace $X$ by a separable topological vector space?



A particular separation example
Suppose that $X$ is a separable Banach space. Then $X^\*$ is weak\*-separable, so take a countable weak\*-dense set $F$ in $X^\*$. Consequently, [$F$ is total](https://math.stackexchange.com/questions/1859334/g-is-dense-in-x-in-weak-sense-if-and-only-if-g-is-total-set), hence the only element on which all elements of $F$ simultaneously vanish is 0.


This works as long as $X$ is a locally convex space and $X^\*$ is weak\*-separable. Note that, by Goldstine's theorem, this will work for every $X=Y^\*$, where $Y$ is a separable Banach space.


