I know that weak* topology is the weakest topology so that $Jx$ is continuous for $\forall x\in X$, where $J$ is the isometry from $X$ to $X''$. But what exactly is this topology? What is the open set in general look like?
And moreover, I want to prove another topology induced by metric is exactly the weak* topology. How can I prove this topology is weaker than weak* topology, so that it is exactly the weak* topology?