In the book Tensor Norms and Operator Ideals by A. Defant and K. Floret, pg. 48, the authors use the fact that the subspace $$Z = \{ T \in \mathcal{L}(X^*,Y) : T\; \text{is weak}^*\text{-weak-continuous} \}$$ is closed in $\mathcal{L}(X^*,Y)$. Here, $X,Y$ are Banach spaces and $\mathcal{L}(X,Y)$ denotes the space of all bounded linear operators from $X$ to $Y$ with the usual norm.
I've tried to prove this but found no success. Can somebody point me to a proof or reference of this fact? Thanks in advance!
If the sequence $\{T_n\}_n\subseteq Z$ converges to $T\in \mathcal L(X^*, Y)$, we must to prove that $T$ lies in $Z$. In order to do this it is enough to show that, for every $y^*\in Y^*$, the linear functional $$ x^*\in X^*\mapsto \langle T(x^*),y^*\rangle \in {\mathbb R} $$ is $w^*$-continuous. This functional is clearly the norm limit of the functionals $$ x^*\in X^*\mapsto \langle T_n(x^*),y^*\rangle \in {\mathbb R} $$ which are $w^*$-continuous since the $T_n$ lie in $Z$ by hypothesis. The conclusion thus follows from the fact that the $w^*$-continuous linear functionals on $X^*$ (aka the cannonical image of $X$) is closed in $X^{**}$.