Does there exist a separable topological vector space $X$ satisfying in the following property?
1- $X^*$ is non separable.
2- For every countable subset $F\subset X^*$ there exists $x_F\in X$ such that $f(x_F)=0$ for all $f\in F$.
Does there exist a separable topological vector space $X$ satisfying in the following property?
1- $X^*$ is non separable.
2- For every countable subset $F\subset X^*$ there exists $x_F\in X$ such that $f(x_F)=0$ for all $f\in F$.