Nontrivial functionals on $l^\infty$ vanishing on $c_0$

Question

I understood that the dual of $c_0$ is a proper subspace of the dual of $l^\infty$, by Hahn-Banach theorem. But how can I find functionals in $(l^\infty)^*$ vanishing on $c_0$?

Answer 1

Following the approach suggested by Daniel Fischer: define $\phi(x) =\lim x_n$ whenever the limit exists. This defines $\phi$ as a bounded linear functional on a linear space containing $c_0$. By definition $\phi$ vanishes on $c_0$. The Hahn-Banach theorem provides an extension of $\phi$ to an element of $\ell_\infty^*$.