I know $H^\infty \subset BMOA \subset H^2$ and $BMOA$ is dual of $H^1$. I know how to prove that $clos_2[H^\infty]=H^2$, here $clos_2$ is the closure in the $H^2$ norm, but I do not know how to prove that the weak star closure of $H^\infty$ in $BMOA$ is $BMOA$.
Can anyone suggest me some references or write down the proof if it is too trivial. Any kind of help would be greatly appriciated. Thanks.