# Boundedness of a Net in Second Dual Transfering to Original Space

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 $$\mathcal B(A, X^{**})$$?