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In functional analysis, Goldstein's theorem says that

For Banach space $X$, $B_X$ is $w^*$-dense(weak-star-dense) in $B_{X^{**}}$.

I want to know whether $B_X, B_{X^{**}}$ can be replaced by $S_X, S_{X^{**}}$.

More specifically, given any $x^{**}\in X^{**}$ with $\Vert x^{**}\Vert=1$, is there a sequence of norm-one elements in $X$ converges to $x^{**}$ in weak-star topology?

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Yes. Let $\{x_\alpha\}$ be a net in $B_X$ that weak*-converges to $x^{**}$. For every $r<1$, the closed ball $rB_{X^{**}} = \{x\in X^{**}:\|x\|\le r\}$ is weak*-closed. Therefore, if $\{x_\alpha\}$ has a subnet staying in such a ball, it cannot converge to $x^{**}\notin rB_{X^{**}}$. In other words, $\|x_\alpha\|\to 1$.

Let $y_\alpha = x_\alpha/\|x_\alpha\|$. Then $x_\alpha-y_\alpha\to 0$ in the norm, hence also in weak* sense. Since $x_\alpha\to x^{**}$ in the weak* sense, it follows that $y_\alpha\to x^{**}$ in the weak* sense.

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  • $\begingroup$ Thanks a lot! I'm sorry to late. I saw this comment just right now T.T $\endgroup$ – CSH Mar 13 '17 at 11:00

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