Condition to separability of a Banach space.

Question

I am trying to prove the following statement:

Let X be a Banach space and $X^{*}$ its topological dual space. If there exists a countable family of functions $(f_{n})_{n} \subset X^{*}$ such that for every $x,y \in X$, $x \neq y$, there exists $m \in \mathbb{N}$ such that $f_{m}(x) \neq f_{m}(y)$ then X is separable.

I don't even know if this is true. I've been looking for counterexamples but the only non-separable Banach space familiar to me is $l_{\infty}$ and I can't find such a family of functions there. Thank you.

Answer 1

Concrete example by Daniel Fischer:

With $f_m(x) = x_m$, you have a countable family in $(l_\infty)^\ast$ such that for all $x\neq y$ there is an $m$ with $f_m(x)\neq f_m(y)$.

Generalization by Tomek Kania:

Actually, there is nothing special about $\ell_\infty$ here. You can take any $X^{**}$ for $X$ separable as a counter-example (this is due to Goldstine's theorem, of course).