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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!

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  • $\begingroup$ Which topology on $Y$ are you considering? $\endgroup$ Commented Mar 21, 2021 at 14:03
  • $\begingroup$ Each $T \in Z$ is continuous from $(X^*, \|\cdot \|)$ to $(Y, \|\cdot\|)$ and continuous from $(X^*, w^*)$ to $(Y,w)$, where $w^*$ and $w$ denote the weak$^*$ and weak topologies, respectively. $\endgroup$ Commented Mar 21, 2021 at 14:08

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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 $X^{**}$.

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  • $\begingroup$ Thank you very much! $\endgroup$ Commented Mar 21, 2021 at 15:49

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