What is the norm of an element in $L^\infty(\mathbb{D})$ using the weak-star topology?

Question

Under the convention that $L^1(\mathbb{D})$ has normalized Lebesgue measure for the unit disc in $\mathbb{C}$, its dual space can be regarded as $L^\infty(\mathbb{D})$. Hence we can equip $L^\infty(\mathbb{D})$ with a weak-star topology. I am trying to gain an elementary understanding of the weak-star topology, and am wondering what the norm is given by in this space. Thank you.