Pointwise convergence in $c_0$ is not equivalent to weak convergence

Pointwise or weak convergence applies to sequences of members of $c_0$, not to individual members of $c_0.$ Let $x^{(m)}=(x_{m,n})_{n\in\Bbb N}\in c_0$ where $x_{m,m}=m^2$ and $x_{m,n}=0$ if $m\ne n$. For each $n$ we have $\lim_{m\to\infty}x_{m,n}=0.$ But $\phi=(1/n^2)_{n\in\Bbb N}\in c_0^*=\ell_1$ and $\phi(x^{(m)})=1$ for every $m.$

If we regard each $x^{(m)}$ as a function with domain $\Bbb N$ (where $x^{(m)}(n)=x_{m,n}$) then $(x^{(m)})_m$ converges to $0$ at each point of the domain $\Bbb N. $But if we regard each $x^{(m)}$ as a function (a functional) with domain $c_0^*$ (where $x^{(m)}(\psi)=\psi(x^{(m)})$ for each $\psi\in c_0^*)$ then $(x^{(m)})_m$ does not converge to $0$ at each point of the domain $c_0^*.$