Let $X,Y$ be two Banach spaces. I'm asked to define the weak-star topology in the set of bounded linear maps from $X$ to $Y^{\prime}$, $B(X,Y^{\prime})$, where $Y^{\prime}$ is the dual space of $Y$. However, I'm not given a normed space $Z$ such that $Z^{\prime}=B(X,Y^{\prime})$. If given, the weak-star topology is the weakest topology such that \begin{align} B(X,Y^{\prime}) \ni M \mapsto Mz \end{align} is continuous for each $z \in Z$. What is $Z$? Or is there an other way to understand the question?
See below:
