Dense convex set in $*$-weak topology

Question

Let $X$ be a Hausdorff topological vector space over $\mathbb{K}$. Suppose $W$ is a convex subset of its topological dual $X'$. How to prove that if for any $x\in X\setminus\{0\}$ set $\{f(x):f\in W\}$ is dense in $\mathbb{K}$ then $\overline{W}^{\sigma(X',X)}=X'$ ?