A bounded subset $B$ of the dual $X^*$ of a Banach space $X$ is called *norming* if the formula $\|x\|:=\sup\{|x^*(x)|:x^*\in B\}$ determines an equivalent norm on $X$.

Observe that the sequence $(e_n^*)_{n\in\omega}\subset c_0^*=\ell_1$ of coordinate functionals in the dual of $c_0$ is norming and weakly$^*$ null.

Question 1.Is it true that every absolutely convex bounded norming set $B\subset c_0^*$ in the dual of the Banach space $c_0$ contains a norming weakly$^*$ null sequence?

The same question can be asked for subspaces of $c_0$.

Question 2.Let $X$ be a closed subspace of the Banach space $c_0$ and $B\subset X^*$ is a norming bounded absolutely convex set. Is it true that $B$ contains a norming weakly$^*$ null sequence?

**Remark.** If a Banach space $X$ admits a norming weakly$^*$ null sequence of functionals $\{f_n\}_{n\in\omega}\subset X^*$, then the operator $T:X\to c_0$, $T:x\mapsto (f_n(x))_{n\in\omega}$, is an isomorphic embedding of $X$ into $c_0$.
Therefore, $X$ is isomorphic to a subspace of the Banach space $c_0$.