Seminorms that define the weak* topology

Let $E$ be a normed space, and $E'$ its dual space. The weak* topology on $E'$ can be defined by the family of seminorms $\{p_x\,|\, x\in E\}$, where for any $f\in E'$, $p_x(f)=|f(x)|$.

My question is, can this topology be defined by fewer seminorms? For example, if we have a countable dense set $\{x_1, x_2, ...\}\subset E$, does the smaller family $\{p_{x_n}\,|n\geq 1\}$ of seminorms define the same topology on $E'$?

One of the difficulties is that open sets in weak* topology are not bounded in the original $\|\cdot\|$ norm.

Solution 1:

For the question in your comment here is a counter-example:

Let $X$ be the space of all finitely non-zero sequences with the sup norm. Its completion is $c_0$ and the dual is $\ell^{1}$. The sequence $(ne_n)$ tends to $0$ in the weak* topology induced by $X$ but not in the weak* topology induced by $c_0$ (since $x_n \to 0$ does not imply $nx_n \to 0$).