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Let $ X $ be a Banach space. Let $ J : X \to X^{**} $ denote the natural embedding of $ X $ into $ X^{**} $.

Suppose that there exists a bounded net $ (T_{\alpha})_{\alpha \in I} \subset X^{**} $ that converges to $ T \in X^{**} $, and let $ M > 0 $ be such that $ \| T_{\alpha} \| \leq M $ for all $ \alpha \in I $.

Is it possible to find a bounded net $ (x_{\beta})_{\beta \in J} \subset X $ (probably related to $ T_{\alpha} $) such that $ J(x_{\beta}) $ weak-* converges to $ T $ and $ \| t_{\beta} \| \leq M $ for all $ \beta \in J $?

Let $ B_X $ and $ B_{X^{**}} $ denote the closed unit balls of $ X $ and $ X^{**} $ respectively. By Goldstine's Theorem, the weak-* closure of $ J(B_X) $ is $ B_{X^{**}} $. For each $ r > 0 $ let $ rB_X = \{ x : \| x \| \leq r \} $ and define $ rB_{X^{**}} $ similarly. As a consequence to Goldstine's Theorem, since $ J $ is an isometry, $ J(rB_X) $ is weak-* dense in $ rB_X^{**} $ for every $ r > 0 $.

Since $ \| T_{\alpha} \| \leq M $ for all $ \alpha $ we see that $ (T_{\alpha}) \subset M B_{X^{**}} $ and also $ T \in M B_{X^{**}} $. Since $ J(MB_X) $ is weak-* dense in $ MB_{X^{**}} $ there exists a net $ (x_{\beta}) \subset MB_X $ such that $ J(x_{\beta}) $ weak-* converges to $ T $.

Is this correct?


Let $ A $ be a Banach algebra and let $ X $ be a normed space. Let $ J : X \to X^{**} $ be the natural embedding of $ X $ into $ X^{**} $. Let $ \mathcal B(A, X) $ denote the space of all bounded linear operators from $ A $ to $ X $, and similarly, let $ \mathcal B(A, X^{**}) $ denote the space of all bounded linear operators from $ A $ to $ X^{**} $.

Let $ (T_{\alpha})_{\alpha \in \Gamma} \subset \mathcal B(A, X^{**}) $ be a bounded net that converges to a bounded linear operator $ T : A \to J(X) $, and let $ M > 0 $ be such that $ \| T_{\alpha} \| \leq M $ for all $ \alpha \in \Gamma $.

Is it possible to find a bounded net of operators $ (t_{\beta}) \subset \mathcal B(A, X) $ such that for each $ a \in A $, $ (J \circ t_{\beta}) $ weak-* converges to $ T $ and that such that $ \| t_{\beta} \| \leq M $ for all $ \beta $?

I am not sure whether this version is true or not, and if it is true, I'm not sure how exactly I can apply Goldstine's Theorem. I must admit, my knowledge on weak-* topologies isn't the greatest. Any help would be greatly appreciated.


Edit: I think my original second question is a bit confusing, so here is what I am really trying to ask: Let $ t \in \mathcal B(A, X) $ and let $ (T_{\alpha}) \subset \mathcal B(A, X^{**}) $ with $ \| T_{\alpha} \| \leq M $ for all $ \alpha $. If $ (T_{\alpha}) $ converges to $ J(t) $ in the strong-operator topology on $ \mathcal B(A, X^{**}) $, then does there exist a net $ (t_{\beta}) \subset \mathcal B(A, X) $ with $ \| t_{\beta} \| \leq M $ for all $ \beta $; such that $ J(t_{\beta}) $ converges to $ J(t) $ in the weak-* operator topology of $ \mathcal B(A, X^{**}) $?

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