# A particular separation example

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?

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, 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.
• Thanks. Any example of separable locally convex space $X$ whose dual is not weak*-separable? Perhaps it serves an answer for Q2! – Ali Bagheri May 11 '18 at 14:48
• $\ell_1$ is separable then its second dual $X=\ell_\infty^{*}$ is weak star separable. But its dual is not separable any more. Right? – Ali Bagheri May 11 '18 at 19:40
• @AliBagheri, certainly $\ell_1^*$ is not norm-separable, however, it is weak*-separable (as is $\ell_1^{**}$). You may be interested in this paper: carma.newcastle.edu.au/brailey/Research_papers/… – Tomek Kania May 12 '18 at 9:09