Solution
= Solution
Take
$$
\boxed{\psi_2(\alpha)=\sum_{n=1}^\infty\alpha_n.}
$$
The 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 $e_n\to0$ in that topology but $\psi_2(e_n)=1$.