Given a locally compact group $G$, consider its Pontryagin dual $G^{'}$. Do compact convergence topology and weak*-topology on $G^{'}$ coincide?. When we talk about weak*-topology on $G^{'}$, it means to view $G^{'}$ to be in the space of $L^{\infty}(G)$ with the relative weak*-topology.
My way is to prove that the identity on $G^{'}$ is a homeomorphism. But I cannot prove the continuity from the weak*-topology to the compact convergence topology. I think $G^{'}$ is not closed in the weak*-topology. Is it right?
I also want to ask that is $G^{'}$ always 1st countable or even metrizable?
Thank you!