 Solution

TakeThe series defines a bounded linear functional of norm one. It corresponds under ${\Phi }_2$ to the constant sequence ${\gamma }_j=1$, so is $\sigma ({\ell }^1,c)$-continuous. Its coefficient sequence is constantly one, which is not in $c_0$, so the continuous-dual result makes it noncontinuous for $\sigma ({\ell }^1,c_0)$. Directly $e_n\to 0$ in that topology but ${\psi }_2(e_n)=1$.